ring theory – louis h rowen a course in ring theory advances in ring theory – j l s chen n q ding advances in ring theory – r v kadison j r ringrose commutative ring t

This chapter contains the general results on rings and their modules which have come to be known as the “structure theory.” One of the main themes is Jacobson’s structure theory,[r]

Ramat Can, Israel

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Library of Congress Cataloging-in-Publication Data Rowen, Louis Halle

Ring theory

(Pure and applied mathematics; v 127-128)

Includes bibliographies and indexes

1 Rings (Algebra) I Title 11 Series: Pure and applied mathematics (Academic Press) ; 127-128 CQA2471

and his book Structure of Rings (1956, rev 1964) remains one of the principal references in the literature (The other classic references are Herstein’s Carus

Monograph Noncommutative Rings and, for the homological point of view, Cartan- Eilenberg’s book Homological Algebra.) However, as more and more

“elementary” proofs found their way into the folklore, the structure of rings began to resemble a tower of Babel, where ring theorists speak such diverse languages (ring theory, module theory, category theory, etc.) that at times we cannot communicate effectively with one another

In the last few years there have been many very successful books in special

topics, which are sometimes used as texts, notably Passman’s book The Algebraic Structure of Group Rings However, any specialized book necessar-

ily presents the subject vertically rather than horizontally; i.e., the selection of topics and proofs reflects the particular needs of the subject area instead of the broader picture, and the reader must often plow through technical results in order to arrive at theorems of more general interest On the other hand, the basic theorems are often proved in an almost offhand fashion, to leave space for the author’s main objective For example, any book concerned with finite dimensional algebras must deal with the Wedderburn- Artin theorem, that any finite dimensional simple algebra is isomorphic to a ring of matrices over a division algebra One of several short, direct proofs is presented, although the reader thereby loses the opportunity of seeing Wedderburn’s theorem in one

xiii

The supplements are designed as a means of continuing a specific thread throughout the text Thus “A Supplement” deals with ordered groups and their uses in ring theory, and occurs in &1.2,1.3; “B Supplement,” dealing with free products, occurs in R1.4, 1.9

The two appendices are akin to supplements, but are put off until the end, in order not to disrupt the flow of the text

The exercises are designed as a secondary extension of the text, and often their “hints” are virtually complete proofs One aim of the exercises is to present interesting theorems (such as the Popescu-Gabriel theorem) which, although important, are not needed later in the text proper With one exception (Bass’ theorem on “big” projective modules in $5.1, which uses Kaplansky’s theorem that non-f.g projective modules over a local ring are free), I do not know of any result in the text whose proof relies on an exercise Also appearing as exercises are several interesting examples dispersed throughout the literature (many due to G Bergman), so the reader is urged at least to peruse the exercises

Foreword xv

Parts of the text are labeled as “digressions.” These are paragraphs that do not bear directly on the remainder of the text and often are given scant ex- planation, in order not to distract the reader from the intended thrust of the exposition The larger digressions point to a major area of recent interest The shorter digressions are asides

Notwithstanding these various attempts to hold down the size of the main text, several worthy subjects have been slighted The original intention was to give a fairly detailed account of the representation theory of Artin algebras, but lack of space required it to be shrunk to the rather meager sketch at the end of Section 2.9 Other areas have been similarly misrepresented

In contrast to the fundamental aspect of Volume I, the goal of Volume I1 is

to give the flavor of the subjects of current research, although in most areas the results fall a few years behind latest work

The reader should also take note of the various indices, which are intended

in part as aids to help organize the material

The issue of acknowledgments is very delicate, since so many people have graciously aided me in this project Ami Braun, P M Cohn, Marie-Paule Malliavin, and Jean-Pierre Tignol have spotted many flaws in earlier versions

of the manuscript Bill Blair gave instrumental advice in bringing Volume I to final form On the other hand, I am always indebted to my mentor Nathan Jacobson, as well as to S A Amitsur Other colleagues and friends who provided useful help and advice include Ephraim Armendariz, Maurice Auslander, Miriam Cohen, S Dahari, Dan Farkas, Ed Formanek, Larry Levy, John McConnell, Chris Robson, Shmuel Rosset, David Saltman, Lance Small, and Robert Snider Thanks are also due to Tony Joseph and Rudolf Rentschler for pointing out the highlights of the theory of enveloping algebra Small’s excellent compilations of reviews (Reoiews in Ring Theory, Amer Math SOC (1980, 1984)) have been an invaluable tool Finally, I would like to thank Klaus Peters of Academic Press for his efforts in bringing the work to fruition

Introduction: An Overview of Ring Theory

In the solar system of ring theory the Sun is certainly the semisimple Artinian ring, which can be defined most quickly as a finite direct product of matrix rings over division rings Much of ring theory is involved in measuring how far

a ring is from being semisimple Artinian, and we shall describe the principal techniques The numbers in parentheses refer to locations in these two volumes

The Structure Theory of Rings

Any simple ring (with 1) having minimal nonzero left ideal is of the form M,,(D),

by the celebrated result known as the Wedderburn- Artin theorem (2.1.25’) The general structure theory begins with Jacobson’s density theorem (2.1.6),

which generalizes the essence of the Wedderburn- Artin theorem to the class

of prirnitioe rings; using subdirect products (52.2) one can then pass to semiprimitive rings

The passage from semiprimitive rings to arbitrary rings leads one to the study of various radicals, which are intrinsically-defined ideals which when 0

make the ring much easier to analyze Thus the radical is the “obstruction” in the structure theory

In the general structure theory there are several radicals, which we consider

in order of decreasing size The biggest, the Jacobson radical (52.5) denoted Jac(R), is 0 iff the ring is semiprimitive Since Jac(R) is often nonzero, one

xvii

xviii Introduction

considers various nilradicals (#2.6), so called because their elements are nilpotent The largest nilradical is called the upper nilradical Nil(R) and the smallest nilradical is called the prime radical or lower nilradical and is the

intersection of the prime ideals; a ring is semiprime iff its lower nilradical is 0

Even in the classical case of finite dimensional algebras over fields (for which

semiprimitive rings become semisimple Artinian (42.3) and for which the

radicals all coincide), the structure of the radical itself is extremely com- plicated; often the radical is considered an encumbrance to be removed as soon as possible Thus the various structure theorems which enable us to

“shrink” the radical take on special significance; most prominent is Amitsur’s

theorem (2.5.23), which says that if Nil(R) = 0 then Jac(R[A]) = 0, where

A is a commuting indeterminate over R Other such results are considered

in 42.6

Certain classes of rings are particularly amenable to the structure theory If

a ring R is Artinian then Jac(R) is nilpotent and R/Jac(R) is semisimple Artinian; furthermore, one can “lift” the idempotents from R/Jac(R) to R to

obtain more explicit information about R ($2.7) There are other instances when Jac(R) is nil (42.5)

One of the principal classes of noncommutative rings is Noetherian rings

(43.5), for which Nil(R) is nilpotent Fortunately for semiprime Noetherian rings one has Goldie’s theorem (43.2), which shows that the classical ring of

fractions exists and is semisimple Artinian; this result makes available the Wedderburn- Artin theorem and thereby rounds out the Wedderburn-

Artin-Noether-Jacobson-Levitzki-Amitsur-Goldie structure theory Goldie’s theorem also applies to prime rings with polynomial identities (Chapter 6) Rings of fractions have been generalized to rather broad classes of rings, as described in Chapter 3

Since the structure theory revolves around primitive and prime rings, one is interested in the set of primitive ideals (of a ring) and the set of prime ideals

These sets are called the primitiue and prime spectra and have a geometry of

considerable interest (42.12) Recent research has focused on the primitive and

prime spectra of certain classes of rings

Since any ring is a homomorphic image of a free ring, the Cohn- Bergman school has conducted research in studying free rings in their own right Such a project is of immense difficulty, for the very reason that the theory of free rings necessarily includes all of ring theory

Occasionally one wants to consider an extra bit of structure for a ring, the

inuolution (42.13), which generalizes the transpose of matrices The structure

theory carries over fairly straightforwardly to rings with involution, with involutory analogues of the various structural notions

Introduction xix The Structure Theory of Modules

Semisimple Artinian rings are also characterized by the property that every module is a direct sum of simple modules (2.4.9), and, furthermore, there are only a finite number of isomorphism classes of simple modules (2.3.13) This raises the hope of studying a ring in terms of its modules, and also of studying

a module in terms of simple modules The “best” modules to study in this sense are Artinian Noetherian modules, for they have composition series ($2.3), and these are essentially “unique.” Unfortunately there is no obvious way of building a module from its simple submodules and homomorphic images, so one turns to building modules as direct sums of indecomposable modules, thereby leading us to the “Krull-Schmidt” theorem (2.9.17), which says that every module with composition series can be written “uniquely” as a direct sum of indecomposables Unfortunately this leaves us with determining the indecomposables (52.9), which is a tremendous project even for Artinian rings and which largely falls outside the scope of this book Thus one is led to try more devious methods of studying modules in terms of simple modules, such

as the noncommutative Krull and Gabriel dimensions (53.5) The Krull di-

mension has become an indispensible tool in the study of Noetherian rings

Category Theory and Homology

When studying rings in terms of their modules, one soon is led to categories of modules and must face the question of when two rings have equivalent cat- egories of modules Fortunately there is a completely satisfactory answer of Morita (54.1)

Every module is a homomorphic image of a free module, and perhaps one could learn more about modules by studying free modules It turns out that a more natural notion from the categorical point of view is projective module

(52.8), and indeed a ring R is semisimple Artinian iff every module is projective (2.1 1.7) The interplay between projective and free is very important, leading to the rank of a projective module (2.12.19) and the K O theory (55.1)

The path of projective modules can take us to projective resolutions and homological dimension of modules (g5.1, 5.2) Homological dimension is most naturally described in terms of category theory, which lends itself to dualization; thus one also gets injective modules (52.10), which also play an important role in the more general theories of fractions (@3.3,3.4) Homology (and cohomology) lay emphasis on two important functors, YU,~ and &xt,

which are derived respectively from the tensor functor and from the functor

Xom These latter two functors are an example of an adjoint pair (4.2)

xx Introduction Special Classes of Rings (Mostly Volume 11)

Certain classes of rings are of special interest and merit intensive research Classically the most important are finite dimensional algebras over a field; these fall inside the theory of Artinian rings, but much more can be said The role of the radical is made explicit by Wedderburn’s principal theorem (2.5.37)

when the base field is perfect; Wedderburn’s result has been recast into the cohomology of algebras (45.3)

Much of the theory of finite dimensional algebras carries over to the more general realm of rings with polynomial identities (PI-rings, chapter 6) Deep results of representation theory can be obtained using the PI-theory by means

of elementary arguments Moreover, PI-theory is not tied to a base field, and

so various generic techniques are available to enrich the PI-theory; relatively free PI-algebras have commanded considerable attention It turns out that there is just enough commutatively in the PI-theory to enable one to obtain

a satisfactory version of much of the theory of commutative algebras and, indeed, to build a noncommutative algebraic geometry (which however lies largely outside the realm of this book)

In a different direction, commutative algebraic geometry leads us to con-

sider a@ne algebras over a field, by definition finitely generated as algebras Although not much can be said about affine algebras in general, affine alge-

bras become more manageable when they have finite Gevand- Kirillov dimen-

sion (46.2); in particular, affine PI-algebras have finite Gelfand-Kirillov di- mension and are a very successful arena for generalizing results from the commutative theory

Returning to the Wedderburn- Artin theorem, a very natural question is,

“What can be said about the division ring D in M,(D)?” This remains one of the more troublesome questions of research today, because tantalizingly little

is known about arbitrary division rings When D is finite dimensional over a field however, a whole world opens up, comprising the theory of (finite di- mensional) central simple algebras and the Brauer group (Chapter 7) Much recent research on division algebras is from the standpoint of noncommu- tative arithmetic and K-theory, but we deal mostly with the general structure theory of division algebras Most of the exposition is given to understanding the algebraic content of several special cases of the amazing Merkurjev- Suslin theorem

Our final chapter (8) is about those rings which arise most in representation theory, namely, group rings and enveloping algebras of Lie algebras These rings in fact contain all the information of the respective representation theories, and therefore provide a key link from “pure” ring theory to the

Introduction xxi

outside world Both areas naturally lie on the border of algebra, drawing also

on analysis and geometry Nevertheless, the ring theory provides a guide to directions of inquiry; the question of central interest in enveloping algebra theory has been to determine the primitive ideals, for these correspond to the irreducible representations Pure ring theory also yields a surprising amount

of information, and much of the theory can be cast in the general framework

of Noetherian rings (58.4) (The structural foundation is laid in 92.12.)

Another topic in modern research is the Galois theory of rings Jacobson developed a Galois theory to study extensions of division rings, in a similar manner to the Galois theory of extensions of fields This has led to the study of

fixed subrings under groups of automorphisms (end of 52.5), and more re-

cently to Hopf algebras, which are treated in $8.4 as a simultaneous gener- alization of group rings, enveloping algebras, and algebraic groups

A word about current research-whereas the 1960s and early 1970s was the era of abstraction and beautiful general theories, the late 1970s and 1980s have displayed a decided return to specific examples Thus considerable recent attention has turned to Weyl algebras and, more generally, rings of differential polynomials, and the theory of the enveloping algebra of sI(2, n) has produced many interesting examples and a few surprises

Table of Principal Notation

Note: ! after page reference means the symbol is used differently in another

part of the text

49 1

0 General Fundamentals

50 Preliminary Foundations

The object of this section is to review basic material, including the fundamental results from ring theory The proofs are sketched, since they can

be found in the standard texts on abstract algebra, such as Jacobson [SSB]

Two important conventions: N denotes the natural numbers including 0; also, given a function f: A + B we shall often use fa instead of the more standard

notation f(a), and similarly fA denotes {fa: a E A} Given sets A, I define

A' = {functions f: I -, A}, which can be identified with the Cartesian product

n { A : i E I } A special case is I = { 1, , n } ; in this case we denote A' as A("),

which can be identified with A x x A (taken n times) under the bijection sendingf to ( f l , ,fn )

Monoids and Groups

A monoid is a semigroup S which has a unit element 1, such that

1s = sl = s for each s in S If a semigroup S lacks a unit element we can ad-

join a formal element 1 to produce a monoid S' = S u { l } by stipulating

1s = sl = s for all s in S

In verifying that a monoid S is a group, one need only check that each element is left invertible, i.e., for each s there is s' such that s's = 1 (Indeed,

1

2 General Fundamentals

take a left inverse s" of s' Then s" = ~"(s's) = (s"s')s = s proving ss' = 1 and

s' is also the right inverse of s, i.e., s' = s-'.)

Sym(n) denotes the group of permutations, i.e., of 1: 1 functions from { 1, , n} to itself; Sym(n) is a submonoid of A A where A = { 1, , n}

Although we have used the multiplicative notation for groups, we shall

usually write abelian groups additively, with + for the group operation, 0 for the unit element, and - g for the inverse of g

Fundamental Theorem of Abelian groups: Eoery finitely generated abelian

group is isomorphic to a finite direct sum of cyclic subgroups

An element of a group is torsion if it has finite order The set of torsion

elements of an abelian group G is obviously a subgroup, and we say G is

torsion-free if its torsion subgroup is 0 Since any cyclic group is isomorphic

either to (Z, +) or some (Z/nZ, +) for some n in Z, we have

Corollary: Eoery finitely generated abelian group is the direct sum of a torsion-fiee abelian group and a j n i t e abelian group

Rings and Modules

A ring is a set R together with operations +, (called addition and multi- plication) and distinguished elements 0 and 1, which satisfy the following properties:

(R, +, 0) is an abelian group

(R, ., 1) is a monoid

a(b + c) = ab + acand (b + c)a = ba + cafor all a,b,cin R

If ab = ba for all a, b in R, we say R is commutatiue, common examples being

Z, Q, R, and C, as well as rings of polynomials over these rings; matrix rings (cf., $1.1) provide examples of noncommutative rings

The singleton (0) is a ring called the trioial ring In all other rings 1 # 0 (since if 1 = 0 then any r = r 1 = r 0 = 0) A subring of a ring is a subset which is itself a ring having the same distinguished elements 0, 1; thus (0) is

not a subring of any nontrivial ring Unless explicitly stated otherwise, rings

will be assumed to be nontrivial

Suppose R, T are rings A ring homomorphism f: R + T is an additive group homomorphism satisfying f(rlr2) = (frl)(fr2) and f l = 1

By domain we mean a (nontrivial) ring in which each product of nonzero

elements is nonzero; R is a division ring (or skew-field) if (R - {O},., 1) is a group, i.e., each nonzero element is invertible A commutative domain is

$0 Preliminary Foundations 3

usually called an integral domain (or entire); a commutative division ring

is called a field

Given a ring R we define a ( l e f ) R-module to be an abelian group M (written

additively), together with composition R x M + M called scalar multipli- cation, satisfying the following laws for all ri in R and xi in M:

r(xl + x 2 ) = rxl + rx,;

( r l r 2 ) x = rl(r2x);

(rl + r2)x = r l x + r,x;

l x = x

In other words, every possible associative law and distributive law involving

scalar multiplication holds A right R-module M is an abelian group with

scalar multiplication M x R + R satisfying the right-handed version of these laws The motivating examples:

(i) R is a left (or right) R-module, where the addition and scalar multiplication (ii) The modules over a field F are precisely the vector spaces over F are taken from the given ring operations of R

Suppose M, N are R-modules A module homomorphism f: M + N , also

called a map, is a group homomorphism “preserving” scalar multiplication in the sense f ( r x ) = rfx for all r in R and x in M In analogy to rings, a map is

an isomorphism if it has an inverse which is also a map; clearly, this is the case

iff the map is bijective However, maps have additional properties Write Hom,(M, N) for {maps from M to N}, made into an Abelian group under

“pointwise” addition of maps, i.e., (f + g)x = f x + gx The zero map sends

every element of M to 0 Moreover, composition of functions provides a map from Hom(N, K ) x Hom(M, N) to Hom(M, K), which is bilinear in the sense

and for any maps h: M + N and gi: N + K

A submodule of an R-module M is an additive subgroup N closed under the given scalar multiplication; in this case N is itself a module and we write

N < M A submodule N is proper if N < M Viewing R as R-module as above,

we say L is a lefi ideal of R if L I R; in other words, a left ideal is an additive subgroup L satisfying rx E L for all r in R and x in L Note that for any x in

a left R-module M we have Rx I M The case M = R has special interest, because a left ideal L is proper iff 1 # L In particular, for r E R we have Rr c R

iff r has no left inverse Right ideals analogously are defined as right sub-

modules of R A is a proper ideal of R (written A 4 R) if A is a proper left and right ideal of R

If f is a ring homomorphism or a module homomorphism, we define its kernel kerf to be the preimage of 0 Then rudimentary group theory shows kerf = 0 iff f is 1: 1 Consequently, f is an isomorphism iff f is onto with

(91 + 92)fl = S l f i + 9 2 f 2 S l ( f 1 + fi) = S l f l + 9 2 f 2

General Fundamentals

4

kerf = 0 Note for a map f : M f N that kerf is a submodule of M and is proper iff f is nonzero; on the other hand, iff: R f T is a ring homomorphism then kerf 4 R

Let us try now to characterize kernels structurally First take R-modules

N I M Forming the abelian group M / N in the usual way, as {cosets of N } ,

we can define scalar multiplication by

r(x + N ) = rx + N for r E R and x E M , thereby making M / N a left R-module called the quotient module (or residue module or factor module) There is a canonical map cp: M + M / N given by

x + x + N , and N = ker cp In this way we see every submodule of M is the

kernel of a suitable map Moreover, we have Noether's isomorphism theorems:

Proposition 0.0.1: Suppose f: M + M' is a map of R-modules whose kernel contains a submodule N of M Then there is a map f: M / N + M' given by

f ( x + N ) = f x , with kerf = (kerf )IN In particular, i f f : M + M' is onto and

kerf = N then f is an isomorphism

Proof(Sketch): These facts are standard for abelian groups, so one need merely check scalar multiplication is preserved Q.E.D

Corollary 0.0.2: If M , , M , I M then MI + M , I M and M , n M , I M , and ( M , + M , ) / M , x M l / ( M l n M,)

Proofi Define f : M , + ( M , + M , ) / M , by f x = x + M a Clearly f is onto and kerf = {x E M , : x + M , = 0 } = M I n M , so apply the proposition

Q.E.D

Corollary 0.0.3: If K I N I M then ( M / K ) / ( N / K ) x M I N

Proofi The canonical map M + M / N yields an onto map M / K + M / N with kernel N I K , so apply the proposition Q.E.D

One also obtains similar results for rings Given 14 R we define the

quotient ring (also called factor ring or residue ring) R/I to have the usual

additive group structure (of cosets) together with multiplication

( r , + I)(r, + I) = r1r2 + 1

This can be easily verified to be a ring, and there is a canonical ring homomorphism rp: R + R / I given by cpr = r + I

$0 Preliminary Foundations 5 Proposition 0.0.4: Suppose f: R + T is a ring homomorphism whose kernel contains an ideal A There is a ring homomorphism f: RIA + T given by

f ( r + A) = f r , and kerf = (kerf )/A If f is onto and kerf = A then 7 is an isomorphism

Corollary 0.0.5: If B E A are proper ideals of R then (A/B) 4 R/B and

(R/B)/(AIB) !z R/A

If f: M + N is a map of R-modules then fM is a submodule of N; thus f

is onto iff N / f M = 0, providing a useful test For rings one sees for any homo- morphism f: R + T that fR is a subring of T Since fR is not an ideal of T

we do not have the parallel test for rings, but anyway we shall find it useful

at times to replace T by fR, thereby making f onto

Algebras

In the text C usually denotes a commutative ring A C-algebra (or algebra over C) is a ringR which is also a C-module whose scalar multiplication satisfies the extra property

c(rlr2) = (crl)r2 = r1(cr2) for all c in C, and r l , r2 in R

Any ring R is also a Z-algebra, by taking nr to be r + * + r (taken n times); the formal correspondence is given more formally in example 0.1.10 below

In general the theories of algebras and of rings are very similar Indeed,

if R is a C-algebra and A 4 R is a ring then A c R as C-module (since

ca = c( l a ) = ( c l ) a E A for all c in C and a in A) Thus the ring RIA also has

a natural C-module structure, with respect to which RIA is in fact a C-algebra Put more succinctly, any ring homomorphic image of R is also naturally

a C-algebra

Define the center of a ring, denoted Z(R), to be (z E R: rz = zr for all r in

R}, clearly a subring of R Under its ring operations R is an algebra over every subring of Z(R) Conversely if R is a C-algebra then there is a canonical ring homomorphism cp: C + Z(R) given by cpc = c l (Proof c l E Z(R) since

(c1)r = cr = c(r1) = r(c1); cp is a ring homomorphism because (clc,)l = cl(c2 1 ) = c l ( l(c2 1 ) = (cl l)(c, l ) ) In case C is a field we have ker cp 4 C so ker cp = 0, and we may identify C with a subring of R Often it is easier to prove theorems about algebras over a field, which one then tries to generalize

to algebras over arbitrary commutative rings

At times we shall need the following generalization of the center of

a ring R Suppose A c R The centralizer of A in R, denoted C,(A), is

6 General Fundamentals

{ r E R:ra = ar for all a in A } , a subring of R We say B centralizes A if

B c CR(A) For example C,(R) = Z(R) Note that A G CR(CR(A))

Proposition 0.0.6: Any maximal commutative subring C of R is its own centralizer

Proofi Let T be the centralizer of C in R Then C G T But for any a in T we

see C and a generate a commutative subring C' of R; by maximality of C we have C' = C so a E C ; hence T = C, as desired Q.E.D

Preorders and Posets

A preorder is a relation which is reflexive (a I a ) and transitive (if a I b and

b I c then a I c) A preorder I on S is called a partial order (or PO for short)

if I is antisymmetric; i.e., a I b and b I a imply a = b In this case (S, I) is

called a poset We write a < b when a I b with a # b The following posets are of particular importance to us

(i) Every set S has the trioial (or discrete) PO defined by declaring any two distinct elements are incomparable (Lea, a I b iff a = b)

(ii) The power set B(A) of a set A is the set of subsets of A, ordered by set

inclusion

(iii) If M is an R-module, define Y(M) = {submodules of M}, partially ordered under I (so N , I N , if N1 is submodule of N,) When R is ambiguous we write 9 ( R M ) for Y(M) One can do the same for {ideals

of a ring}

Upper and Lower Bounds

Suppose S is a set with preorder I An upper bound for a subset S' of S is an element s in S for which s' I s for all s' in S' Upper bounds need not exist For example, no pair of distinct elements has an upper bound if I is the trivial

PO On the other hand, we call S directed (by I) if every pair of elements has

an upper bound An upper bound s of S' is called a supremum if s I s' for every upper bound s' of S' Dually, we define lower bound and say S is directed from below if every pair of elements has a lower bound A lower bound s of

S' is called an injrnum if s 2 s' for every lower bound s' of S' The supremum (resp infimum) is denoted as v (resp A ) The supremum and infimum of S (if they exist) are denoted, respectively, as 1 and 0

For any poset (S, I) we can define the dual poset (S, 2 ) by reversing the inequality, i.e., now s1 2 s2 if previously s1 I s, The technique of passing to

$0 Preliminary Foundations 7

the dual poset is extremely useful and often produces extra theorems with no extra work

Lattices

A lattice is a poset in which every pair of elements has both a supremum and

an infimum; the lattice is complete if every subset has both a supremum and

an infimum Passing to the dual reverses v and A , so we see the dual of a (complete lattice is also a (complete) lattice The following posets are in fact complete lattices:

(i) The power set Y(A), where v is set-theoretic union and A is intersection

(ii) 9 ( M ) for an R-module M; here v Mi = M i and A M i = n Mi In fact,

this important example is the raison d'itre for the study of lattice theory

by ring theorists, and many purely lattice-theoretic results will have applications throughout the text (cf., exercises 3- 13)

This language enables us to state certain module-theoretic results more

sharply To see this we need a useful general result Given posets A , B we

say a function f: A -+ B is order-preseruing if a , < a, implies fa, < fa, for all a1,a2 in A If A, B are lattices, we say a function f: A -+ B is a lattice homomorphism if f(a, v a,) = fa, v fa, and f ( a , A a,) = fa, A fa, for all

a,, a, in A Every lattice homomorphism f: A -+ B is order-preserving, since

if a, I a, we have fa, = f(a, v a,) = fa, v fa,

Proposition 0.0.7

isomorphism iff f and f -' are order-preserving

Suppose f: A -+ B is a lattice bijection Then f is a lattice

Proof: (a) as observed above ( G ) For any a1,a2 in A we have fa, I

f(al v a z ) and fa, I f(al v a,) so fa, v fa, I f(al v a,) Applying this argument to f - ' yields

a, v a, = f-'fa, v f-lfu, I f-'(fa, v fa,) I f-'f(a, v a,) = a, v a,

so equality holds at each stage The proof for A is analogous Q.E.D

We can apply this result to the lattice 9 ( M )

Proposition 0.0.8: Iff: M -+ N is onto then there is a lattice isomorphism from

9 ( N ) to {submodules of M containing kerf}, given by N ' -+f-'N' (The

inverse correspondence is given by M' -+ fM')

Proof: Group theory yields us a 1: 1 correspondence from {subgroups of N}

to {subgroups of M containing kerf}, and we wish to restrict this

8 General Fundamentals

to submodules If N‘ I N then f -IN’ I M since f (rx) = rfx E N’ for all r

in R and x in f -IN‘; if M‘ I M then clearly f M ‘ I f M = N Thus

we have an order isomorphism, which by proposition 0.0.7 is a lattice isomorphism Q.E.D

The corresponding result for rings (proposition 0.0.10) can be obtained directly, but for the sake of variety one could make use of the following observation

Remark 0.0.9: (“Change of rings”) Suppose f:R + T is a ring homomor- phism Any T-module M can be viewed as R-module via the scalar multipli- cation defining rx to be ( f r ) x for r in R and x in M

Thus every submodule of M in T - d u d can be viewed in R - d u d , providing a lattice morphism from 9 ( , M ) to U ( , M ) When f is onto this

lattice morphism also is onto, since any R-submodule of M is also a T-submodule under the corresponding action In particular, viewing T naturally as T-module we can identify {left ideals of T} = 9(,T) with 9 ( , T )

Proposition 0.0.10: Any onto ring homomorphism f: R + T induces a lattice

isomorphism of {left ideals of R containing kerf} with {left ideals of T};

likewise for right ideals In particular kerf Q R, and f induces a lattice iso-

morphism of {ideals of R containing kerf} and {ideals of T}; the inverse

correspondences each are given by f -I

Proofi Proposition 0.0.8 and remark 0.0.9 yield the following composition

of lattice isomorphisms:

{R-submodules of R containing kerf } + LQRT) + 9(,T)

The rest is clear Q.E.D

Modular Lattices

Since proposition 0.0.8 concerns lattices (of submodules) one is led to look for a more lattice-theoretic approach to modules, which stems from the following observation:

Remark 0.0.11: In any lattice 9 we have a v (b, A b,) I a v b, for i = 1,2,

so a v (b, A b,) I (a v b,) A (a v b,) In particular, if a I b we have

a v (b A c) I b A (a v c)

80 Preliminary Foundations 9

In certain cases equality actually holds The most important case is the

lattice of submodules Y ( M ) , for if MI I M, and x, E M, n (MI + M 3 )

then x, = x1 + x3 for xi in Mi so x3 = x, - x1 E M,, proving x, E M, + (M, n M 3 ) We are led to the following definition:

Definition 0.0.12:

a s b i n 9

A lattice 9 is modular if a v (b A c) = b A (a v c) for all

9 ( M ) is thus a modular lattice A more symmetric condition in verifying the modularity of a lattice is given in exercise 2

Definition 0.0.13:

complement a' with a A a' = 0 and a v a' = 1

A lattice 9 is complemented if for each a in Y there is a

The lattice 9 ( S ) is complemented On the other hand, our other major

example of a lattice 9 ( M ) is in general not complemented Complements play

an important role in module theory because of exercises 12 and 13, which anticipate key structural results in 42.4

Definition 0.0.14: A jilter of a lattice 9 is a subset 9 satisfying the follow- ing three conditions:

(i) If a E F and b 2 a then b E 9

Then {complements of finite subsets of S ) is a filter, called the Frechet

filter, or cojinite filter

A direct way of obtaining a filter 9 of a lattice Y is by finding B E 9

satisfying properties (ii) and (iii) of definition 0.0.14, i.e., 0 $ B and if b, b' E a

then b A b' 2 b" for some b" in B Then {a E 9: a 2 b for some b in B } is a

filter 9, and a is called the base of the jilter 9 We shall define many filters

via their bases

10 General Fundamentals

Zorn ’s Lemma

A poset (S, I) is a chain if for all sl, s, in S we have s1 I s, or s, I s,; in this case I is called a total order; for example, (Z, I) is a chain under the usual (total) order Given a lattice (S, I), one sometimes finds that I induces a total order on a certain subset of S For example, if A, E A, E A, E are subsets of A then {A,: i E N} is a chain in (B(A), E) This situation is of great

interest because of the following result, often called Zorn’s lemma We say a

poset (S, I) is inductive if every chain S’ in S has an upper bound in S For example, (N, I) is not inductive since there exist chains not bounded from above, but (N, 2 ) is inductive since 0 is an upper bound The real interval

[0,1] is inductive with respect to I, but Q n [O,a] is not inductive An element s of a poset S is maximal if there is no element s‘ > s in S Minimal elements are defined analogously (Note that S may have many distinct maximal elements, the most extreme example being when the PO is trivial,

in which case every element is both maximal and minimal

(“Zorn’s lemma”):

maximal element

If (S, I) is an inductive poset then S has at least one

The key application is as follows: Suppose A is a set and S c @ ( A ) such that for any chain {A,: i E I} in S we have u A, E S Then some subset of A is maximal in S

The most basic application of the maximal principle in ring theory is that every proper ideal of a ring R is contained in a maximal proper ideal, called

a maximal ideal Indeed, an ideal A is proper iff 1 $ A, so {proper ideals

of R} is inductive Likewise any (proper) left ideal is contained in a max- imal left ideal Incidentally this does not hold in general if we do not stipu- late the existence of the element 1, which is the main reason we deal with rings with 1, cf., exercise 14

However, a ring need not have minimal nonzero left ideals (for example E),

since the above argument has no analogue

Remark 0.0.26: Any filter is contained in an ultrafilter by the maximal principle

Remark 0.0.27: Using remark 0.0.16 one sees that a filter 9 of a com- plemented lattice is an ultrafilter iff for all a in 9, either a E 9 or a‘ E 9 One concludes that if a v b is in an ultrafilter 9 then a E 9 or b E 9?

Another important application of the maximal principle: {commutative subrings of R containing Z ( R ) } is inductive and thus has maximal members

50 Preliminary Foundations 11

In other words, any ring R has maximal commutative subrings, and these are

often useful in the study of R, in view of proposition 0.0.6

The maximal principle is proved by drawing from set theory and, in fact, is

equivalent to the axiom of choice, which asserts that for any family {Si: i E I }

of sets there is a suitable “choice” function f: I + u Si with f i E Si for each i, i.e., f “chooses” one element from each Si At first blush this axiom seems

obvious; however, the larger cardinality the index set I, the less credible the

axiom becomes P J Cohen proved that the axiom of choice is independent

of the Zermelo-Fraenkel axioms of set theory, and today it is used freely by algebraists because the maximal principle is so powerful To understand the connection we must bring in transfinite induction

Well- Ordered Sets and Transjinite Induction

Many definitions in general ring theory rely on transfinite induction To

understand this process requires some intimacy with the ordinals, and to this end we bring in some formalism from set theory It is natural to build sets from the bottom up, starting with the empty set and then building sets whose elements themselves are sets Thus we formally define the symbols

0 = @, i = 0 u (0) = {@}, Z = i u {i} = {a, {a}},

and so forth The axiom of regularity states for every set S # 0 there is s E S

with s n S = 0 This ensures that given two sets S,, S, we cannot have both

S, E S, and S , E S , (Indeed take S = {Sl, S,}.) Thus we can define an anti- symmetric relation < on sets by

S, < S , whenever S, E S, or SI = S,

Since the elements of a set S are themselves sets, we can view I as an anti- symmetric relation on the elements of S S is called an ordinal if (S, I) is

a chain

If a is an ordinal then a+ = a u { a } is also clearly an ordinal, called the

successor of a In particular 0 = 0, = 0+, 2 = i+, are all ordinals On the other hand, there are ordinals which are not successors, the first of which

is {E: n E N}; these are called limit ordinals

A chain is well-ordered if every nonempty subset has a minimal element

Every ordinal a is well-ordered under < as defined above; indeed if @ #

S c a then any s E S with s n S = 0 is minimal in S Thus we have the fol-

lowing generalization of mathematical induction:

Principle of transfinite induction: Suppose a is an ordinal and S c a has the property for every ordinal a‘ < a that if {fl:fi < a’} G S then a’ E S Then

S = a

12 General Fundamentals

The proof is rather easy, but the applications are wide-ranging; here are some set-theoretic implications we shall need (cf., exercise 15-20):

(i) Every set can be put into 1:l correspondence with a suitable ordinal and

thus is well-ordered under the corresponding total ordering (Thus we shall often describe a set S as {s1,s2, .} even when S is uncountable); (ii) Zorn’s lemma, as stated above

Fields

In the structure theory of rings one often considers fields as “trivial” since they have no proper ideals # 0; in fact, most results from ring theory hardly require any knowledge of fields Nevertheless, fields do play important roles

in several key topics (such as division rings), and ideas from field theory pro- vide guidelines for generalization to arbitrary rings When appropriate we shall assume familiarity with the Galois theory of finite dimensional field extensions, including normal and separable extensions, and the algebraic closure of a field

First Order Logic

We shall need some tools from formal logic, in order to deal with certain important constructions, most notably reduced products and ultraproducts The reader willing to accept these results (1.4.1lff) might well skip this ac-

count On the other hand, some rigor is sacrificed A j r s t order language 9

consists of

constants (or object symbols) which are “designated elements” in the theory variables (or dummy symbols) denoting “variables” or “indeterminates”

relative symbols of order n for n = 0, 1,

function symbols of order n for n = 0, 1,

connectives which are the symbols -I, A , v , denoting, respectively, “not,”

quantijiers which are the symbols V (denoting “for all”) and 3 (denoting

A term is an expression defined inductively as follows: Every constant

or variable is a term, and f(tl, ., t,,) is a term for all terms t l , , t,, and every n-ary function symbol f An atomic formula is an expression without

$0.1 Categories of Rings and Modules 13

Formulas and the free oariables of a formula cp (denoted FV(cp)) are de- fined inductively in terms of rank If cp is atomic then we say rank(cp) = 0 and FV(cp) is the set of variables used in writing cp Inductively, if cpi are formulas

of rank n, then

i cp, is a formula of rank n, + 1 whose free variables a;e FV(cp,)

(cp, A cp,), (cp, v cp,), each is a formula of rank n, + n, + 1 whose free

(Vx)cp, and (3x)cpl are formulas of rank n, + 1 for any x in FV(cp,), whose variables are FV(cp,) u FV(cp,)

free variables are FV(cp,) - {x}

A jirst-order (or elementary) sentence of 8 is a formula without any free variables

A structure Y is a set S together with an assignment of the symbols in 9

to their interpretations in S Constants are assigned to designated elements

of S, relative symbols are assigned to relations on S, n-ary function symbols

are assigned to functions S(”) * S, and variables denote arbitrary elements

of S Let us define when a sentence cp “holds” in 9 Suppose S is the under- lying set

(i) The atomic sentence ( t , = t , ) holds in Y iff the terms t , and t , have the same assignment in S The atomic sentence R ( t , , , t,) holds in S iff the relation R is satisfied by the assignments of t , , , t,

We proceed inductively on rank, having just handled the rank 0 case (ii) i cp holds (in 9) iff cp does not hold (in Y)

(iii) cpl A 41, holds iff cp,, cp, both hold

cpl v cp, holds iff cp, and/or cp2 hold

(iv) (Vx)cp(x) holds iff cp(s) holds for all elements s in S; (3x)cp(x) holds iff cp(s)

holds for some element s in S Here ~ ( s ) denotes the formula obtained by replacing each occurrence of x by s and is easily seen to be a sentence Certain redundancies in the language 8 can be discarded when proving general assertions about the lower predicate calculus In particular, the quantifier 3 and the connective v are superfluous, as the reader can readily check

80.1 Categories of Rings and Modules

The language of categories is useful, particularly in certain aspects of

module theory We presuppose a nodding acquaintance with this language; Jacobson [80B, Chapter 13 more than suffices for this purpose In particular,

the reader should know the definition of category, subcategory, (covariant)

d’e is a subcategory of 92, the category of sets Often the class of objects

of a category is “too large” to be a set, for one can construct distinct objects for each ordinal Such is the case with Wimg and R - A d , cf., exercise 1.4.1

Accordingly, a category is called small when its class of objects is a set The

next example is quite useful

Example 0.1.1: Any set 1 with preorder I can be made into a small cate- gory whose objects are elements of 1, with Hom(i, j) either a singleton or 0

depending on whether or not i I j Write E: for the morphism from i to j if

i 5 j Then E : = l i and E: = E ~ E ; for all i, j, k in I Conversely, any small cate-

gory in which JHom(A,B)) I 1 for all objects A, B can be given a preorder (as a set) and can be described as above

If 1 is an arbitrary set then it can be given the trivial preorder, in which case the corresponding small category has Hom(i, j) empty unless i = j, in which case Hom(i, i) = l i This trivial example is useful in understanding certain constructions of Chapter 1 (e.g., the coproduct)

Definition 0.1.2: Given a category V we define the dual category WoP by

Ob Vop = Ob W and Hornrpop(A, B) = Hom,(B, A) with composition in Cop

given by g f = fg (where f E Homro,(A, B) and g E Homro,(B, C)) In other words, we reverse arrows and write things backwards (In particular

(VO”)”P = W.)

Any general theorem for all categories a fortiori holds for the dual cate- gories; translating back to the original category yields a new theorem, the

dual theorem obtained by switching all arrows The main problem with this

approach is that few theorems hold for all categories, and the dual of a well- known category may be quite bizarre Nevertheless, there are certain impor- tant examples for which the dual is well-known and useful For example, the dual of example 0.1.1 corresponds to the reverse preorder on S, justifying the customary use of “dual” concerning posets and lattices

50.1 Categories of Rings and Modules 15

Monics and Epics

Recall a morphism f: A + B is monk if f g # f h for any g # h in Hom(C, A),

for all objects C; dually f is epic if gf # hf for any g # h in Hom(B, C) Clearly the composition of monics (resp epics) is monic (resp epic) In any sub- category of Yet! each 1:l morphism is monic, and each onto morphism is

epic; we would like to test the converse for R - A d and Wiltg

Proposition 0.1.3: In R - A d , monics are 1: 1, and epics are onto

Pro08 Suppose we are given f: M + N If f is monic define g, h: kerf + M

by taking g to be the identity and h = 0; then f g = f h = 0 implying g = h,

so kerf = 0 and f is 1:l If f is epic then define g, h: N + N / f M by g = 0 and hy = y + f M ; then gf = hf = 0, implying g = h so N / f M = 0, i.e.,

f M = N Q.E.D

The story ends differently for W i ~ g

Example 0.1.4: A 1:l ring homomorphism which is epic and monic, but

not onto Consider the 1 : 1 ring homomorphism f: Z + Q given by f n = n

For any morphism g : Q + R in W i ~ g we have g(mn-’) = (gm)(gn)-’ for all

rn, n # 0 in Z, implying g is determined by its restriction to Z; it follows at once that if g # h then g f # hf, so f is epic

In Exercise 1 we see that all monics in Wilt9 are 1:l Nevertheless, exam-

Perhaps rings categorically should be viewed in terms of the following ple 0.1.4 could be called the tragedy of Wiltg

example

Example 0.1.5: Suppose R is a ring We form a category with only one

object, denoted A, and formally define Hom(A,A) = R where the composi- tion of morphisms is merely the ring multiplication

To differentiate the approach to rings and to modules, we designate ring homomorphisms as “homomorphisms;” a 1 : 1 (ring) homomorphism is

called an injection, and an onto homomorphism is a surjection For modules

we adopt, respectively, the more categorical terminology of map, monic, and

epic Nevertheless, since an ideal of R is merely a left and right submodule,

we would like to introduce another category

Definition 0.1.6: Suppose R, R’ are rings An R-R’ bimodule is a left R-

module M which is also a right R’-module satisfying the associativity

16 General Fundnmentals

condition (rx)r’ = r(xr‘) for all r in R, x in My and r’ in R’ R-Mud’-R‘ is

the category whose objects are R-R‘ bimodules and whose morphisms

f: M + M’ are maps both in R-Mod and d u d - R ‘

Remark 0.1.7: The R-R sub-bimodules of R are precisely the ideals of R

defined by f ’ h = hf for each h in Horn@, A,)

Other functors we need are the identity functor l,, and the “forgetful

functors.” A subcategory W of 9 is full if Hom,(A,B) = Hom,(A,B) for any objects A, B in V Given a full subcategory V of 9 we say the inclusion functor F: W + 9 has a retraction G: .9 + %f if G F = 1,

(kerf)M = 0 On the other hand, if M E R-Mud and (kerf)M = 0 then we

can reverse the procedure and view M as T-module by putting ( f r ) x to be rx

In this way F yields an isomorphism from T-Mud to the full subcategory W

of R-Aod consisting of those modules M such that (ker j ) M = 0 There is

a retraction G: R - A d + V given by GM = M/(ker f )M

$0.1 Categories of Rings and Modules 17

Two categories V, 9 are isomorphic if there exist functors F : V + 9 and

G : 9 * V with G F = 1% and F G = la This definition is very stringent, but

is useful in identifying a pair of theories

Example 0.1.10: d& and Z - A u d are isomorphic categories (Indeed, let F: Z - A u d + d& be the forgetful functor, and define G: d& + Z - A d as follows: Given M E d& we view M as Z-module by introducing scalar multi- plication nx = x + * + x, the sum taken n times for n E N, and (- n)x =

-(nx) Every group homomorphism then becomes a morphism in Z - A d ,

so we have the inverse morphism to F.)

To identify categories of left and right modules, we need the notion of the

opposite ring RoP

Definition 0.1.11: If R is a ring, RoP is the ring obtained by keeping the same additive structure but reversing the order of multiplication, (i.e., the product of rl and r2 in RoP is r2r1)

RoP yields the dual category of the category obtained from R in exam- ple 0.1.5, thereby justifying the notation Note (R"P)"P = R

Proposition 0.1.12: R-Aod and Aud-RoP are isomorphic categories Proofi Given M in R-Aod, we define the scalar product of x E M and

r E RoP to be rx; reversing the order of all products, we see this new scalar multiplication makes M a right R"P-module Any morphism f: M , + M 2 in

R - A d can be viewed at once as a morphism in Aod-RoP, so we thus have

a functor R-Aud+.Mud-RoP We have an analogous functor Aud-R"P+ (RoP)"P-Aod = R-Mud, and the composition of these two functors is the identity Q.E.D

Thus any general theorem about modules is equivalent to a correspond- ing theorem about right modules Usually we want to weaken the notion of

isomorphism of categories to equioalent categories, c.f., Jacobson [80B,

p 271; we shall see in Chapter 4 that categorical equivalence is a funda- mental tool of module theory We shall also need the following construc- tion from time to time

Example 0.1.13: Suppose I is a small category and V is a category Then there is a category V' whose objects are the functors from I to %? and whose

18 General Fundamentals

morphisms are the natural transformations, i.e., Hom(F, G) = {natural trans- formations from the functor F to the functor G} The composition cq of two natural transformations q: F + G and r: G f H is given by (cn), = 4‘,qA E

Hom(FA, HA) for each A in Ob %?

90.2 Finitely Generated Modules, Simple Modules,

and Noetherian and Artinian Modules

Returning to modules, we approach one of the nerve centers of the sub- ject and look at the generation of modules by elements Finitely generated (fag.) modules, also frequently called “finite” and “of finite type,” turn out to

be much more tractable than arbitrary modules In particular, we shall ex- amine cyclic modules, leading us to simple modules At the end we introduce the important classes of Artinian ’modules and Noetherian modules

Finitely Generated Modules

Given M in R d o d and AGR, SGM, define A S = ~ : , , a , s , : t ~ N , a+A,

si E S}; if M E A u d - R we define SA analogously For S = {s,: i E I } we often write Asi instead of AS Usually A will be an additive subgroup of R,

in which case AS is a subgroup of M for any set S; in fact As = {as:a E A}

Proposition 0.2.1: Suppose SG M and M E R - A u ~ If L I R then L S I M

RS is the intersection of all submodules of M which contain S If M E R-

Mod-R’ then RSR’ is the intersection of all sub-bimodules of M containing S

Proofi LS is an additive subgroup of M, and for any r in R, r(LS) =

(rL)S E LS proving LS 5 M In particular RS I M Now for any N I M with S G N we have RS E RN E N; since RS itself is a submodule contain- ing each element s = 1s of S we get the second assertion The last assertion is

proved similarly Q.E.D

We say a module M is spanned by the subset S if M = RS A module

spanned by a finite set is called Jinitely generated, abbreviated as “f.g.”

throughout the text

Definition 0.2.2: R - F ~ M u ~ is the full subcategory of R - A d whose objects are the f.g R-modules

Remark 0.2.3: If f: M + N is a map of modules and M is spanned by S

then fM is spanned by fS (for if x = E l l s i then f x = c r i ( fsi)) In par-

90.2 Finitely Generated, Simple, Noetherinn and Artininn Modules 19

ticular, if M is f.g then f M is f.g On the other hand, a submodule of an f.g

module need not be f.g as seen in the next example

Example 0.2.4: Let Q[A] be the ring of polynomials in one commuting indeterminate over Q and let I = 3,Q[3,] and R = Z + I c A Then Id R but

1 4 R-Firnod (Indeed, given any xl, , x, in I , let the coefficient of 3, in xi

be mi/ni and observe (2n, *n,)-'3, E I - Rxi.)

Cyclic Modules

Of particular interest are modules spanned by a single element

Definition 0.2.5: M is a cyclic R-module if M = Rx for some x in M

R = R1 is a cyclic R-module, so remark 0.2.3 shows RIL is cyclic for every

L < R Conversely, every cyclic module has this form, as we shall see shortly

Dejnition 0.2.6: If M E R-Mud and S c M , define Ann,S (the left anni- hilator of S in R) to be {r E R: rs = 0}, a proper left ideal of R If R is under- stood, we write Ann S for Ann, S; we also write Ann x for Ann{x}

Lemma 0.2.7: If M E R-Mod, then for every x in M there is a map f,: R + M given by f,r = rx; kerf, = Annx, implying RIAnnx w Rx 5 M

Proof: Clearly f, is a map, and kerf, = {r E R: rx = 0} = Ann x, so Rx =

f x R w R/ker f , = R/Annx Q.E.D

Proposition 0.2.8: M E R-Mod is cyclic iff M w R / L for some left ideal L

of R; in fact, for M = Rx then we can take L = Annx

Proof: As noted above, RIL is cyclic; the converse is lemma 0.2.7 Q.E.D

Simple Rings and Modules

We shall turn now to a basic philosophy concerning arbitrary categories One should like to examine objects by taking morphisms to objects whose struc- ture we already know Then the simplest objects would be those objects from which all morphisms are monic, motivating the following definition

Definition 0.2.9: A nonzero module M is simple if M has no proper non-

zero submodules; a ringR is simple if R has no proper nonzero ideals (Simple modules are called irreducible in the older literature.)

20 General Fundamentals

Remark 0.2.10: In WiBg or in R-~?ud, an object A is simple iff every non- zero morphism f: A + B is 1: 1 (Proof: (a) kerf # A so kerf = 0 (e) Any morphism A + A/Z is 1 : 1 so I = 0.)

In particular, we have the following categorical criterion for a module M

to be simple: Every nonzero map from M is monic An analogous criterion

holds for W i ~ f in view of exercise 0.1.1, but the dual criterion only works for modules

Remark 0.2.11: An R-module N is simple iff every nonzero map f: M + N

is onto (Proof: (a) 0 # jM < N implies f M = N; (e) if 0 # N' < N then

the injection N' + N is not onto.)

There are two immediate difficulties in trying to build a structure theory based on simple rings and/or modules:

(i) There must be enough simples to yield general information about rings (ii) One needs some technique to study the simples

The first difficulty can be dealt with by means of maximal left ideals

Remark 0.2.12:

and modules

(i) If L is a maximal left ideal of R then R/L is a simple R-module (Immediate from the lattice correspondences pertaining to R as R-module.) (ii) If I is a maximal ideal of R then R/I is a simple ring

Surprisingly this remark provides all the simple modules

Lemma 0.2.13: If M is a simple R-module then M is cyclic In fact Rx = M for eoery x # 0 in M

Proof: 0 # Rx I M so Rx = M Q.E.D

Proposition 0.2.14: M E R-Aud is simple iff M x R/L for a suitable maxi-

mal left ideal of R

Proof: (=.) Write M = Rx and define cp: R + M by cpr = rx Then cp

is onto so M x Rlkercp, and kercp is a maximal submodule by proposi- tion 0.0.8 (.=) Reverse the argument Q.E.D

Before putting this idea aside, we note an important generalization

50.2 Finitely Generated, Simple, Noetherian and Artinian Modules 21 Proposition 0.2.15: If M E R-.Fimod and N < M then N is contained in some maximal submodule M’ Consequently M J M ’ is a simple module in which the image of N is 0

Proof; Write M =I:=] Rxi, where all x i € M For any chain M , I M , I

of proper submodules, some xi # Mj for all j (since otherwise all xi E Mj for

large enough j, implying M = z R x i E Mj contrary to Mi proper) Thus

xi 4 u M j , which hence must be proper (and is clearly a submodule) Taking

M I = N , we have proved {proper submodules of M which contain N } is inductive and thus by Zorn’s lemma contains maximal members, which must

be maximal submodules of M The rest is clear Q.E.D

It is difficult to study simple rings without imposing further restrictions

because of the intrinsic complexity of ideals Indeed, the smallest ideal of R containing a given element r is RrR = { c: = ril rri2 : t E N, ril, ri2 E R } which

cannot be described in the first-order theory of rings On the other hand, the

smallest left ideal containing r is Rr = {r’r: r’ E R } , which is much more

amenable When r is in the center then in fact RrR = Rr, and the situation is

much easier to handle In general, the classification of simple rings is an immense project, far from completion, but there is the following easy result concerning rings without proper left ideals

Proposition 0.2.16:

(i) Rr = R iff r has a left inverse in R ;

(ii) R is a division ring iff R has no proper nonzero left ideals;

(iii) if R is simple then Z ( R ) is a Jield

Proof;

(i) Rr = R iff 1 E Rr iff 1 = r’r for some r’ in R

(ii) (3) follows at once from (i) Conversely if 0 # r E R then Rr = R

proving every element of R-{0} has a left inverse, so R-(0) is a multiplicative

A poset (S, I) satisfies the maximum (resp minimum) condition if every non-

empty subset has a maximal (resp minimal) element (S, I) satisfies the

22 General Fundamentals

ascending chain condition (abbreviated ACC) if there is no infinite chain s1 < s2 c s3 < * , i.e., if every ascending chain is finite; dually (S, I) satis-

fies DCC if every descending chain is finite

Proposition 0.2.17: A poset (S, I ) satisfies the maximum condition iff it satisfies ACC (S, I) satisfies the minimum condition ifl it satisfies DCC

Proofi We prove the first assertion; the second is its dual and follows by passing to the dual poset First note that an ascending chain is finite iff it contains a maximal element So if (S, I) satisfies the maximum condition then every ascending chain is finite, proving (S, I) satisfies ACC Con-

versely, if (S, I) satisfies ACC then every subset s' is inductive (because every

chain of S' is finite) and thus has a maximal element Q.E.D

Thus to verify a chain is well-ordered, we need only check that there is no

infinite descending subchain Also we see that if a lattice satisfies DCC then

every subset is well-ordered

Noetherian and Artinian Modules

The point of studying chain conditions on lattices is in utilizing the lattice

Y ( M ) of submodules

Defiition 0.2.18: A module M is Noetherian if Y ( M ) satisfies ACC or,

equivalently, if Y ( M ) satisfies the maximum condition M is Artinian if

Y ( M ) satisfies DCC or equivalently the minimum condition

Proposition 0.2.19: Suppose M E R-Aud and N I M M is Noetherian

iff N and M I N are Noetherian M is Artinian iff N and M / N are Artinian Proofi This is really a fact about ACC and DCC in modular lattices hav-

ing 0 and 1, i.e., Y ( M ) satisfies ACC (resp DCC) iff Y ( N ) and Y ( M / N )

satisfy ACC (resp DCC) (*) ACC and DCC certainly pass to sublattices

But Y ( N ) I Y(M) in the obvious way, and Y ( M / N ) is isomorphic to

the sublattice {submodules of M containing N ) by the correspondence

80.3 Abstract Dependence 23

in 9 ( N ) and Y ( M / N ) , respectively; so for some m we have Mi n N =

Mi+1 n N and (Mi + N ) / N = ( M i + , + N ) / N for all i 2 m Thus we need to show that if Mi n N = Mi+ , n N and (Mi + N ) / N = (Mi+ , + N ) / N then

M i = M i + l Suppose X E M ~ + ~ Then X E M , + ~ + N = M , + N ; writing

x = y i + x ‘ for yi in Mi and X ’ E N we have x ’ = x - y , ~ M ~ + ~ n N = M , n N

so x E Mi as desired Q.E.D

Corollary 0.2.21: Every f.g module over a left Noetherian ring R is Artinian) then M is Noetherian)

Proofi We prove Noetherian; Artinian is analogous Let N = 1:;: 4,

which is Noetherian by induction on t But M / N =(N + N,)/N w N,/(NnN,)

is Noetherian so M is Noetherian by the proposition Q.E.D

We say R is left Noetherian (resp left Artinian) if R is Noetherian (resp

Artician) as R-module

Corollary 0.2.21: Every f.g module over a left Noetherian ring R is Noetherian Every f.g module over a lejl Artinian ring R is Artinian

Proofi We prove Noetherian Write M =xi= Rx, Each Rxiw R/Annxi

is Noetherian, so M is Noetherian Q.E.D

80.3 Abstract Dependence

This discussion is motivated by the observation that modifying the proof in Herstein [64B, Ch.4, corollary 23 shows that all bases of a given vector space over a division ring have the same cardinality

Definition 0.3.1: A strong dependence relation on a set A is a relation Edcp

between elements and subsets of A, satisfying the following axioms for all

S,S’ c A:

(i) If s E S then s E~~~ S

(ii) If x Edep S then x Edep S, for some finite S , c S

(iii) If x edep S and s Edep S’ for all s in S then x Edep S’

(iv) If y Edep S u { x } and y $dep S then x Edep S u { y}

The fourth axiom, called the (Steinitz) exchange axiom, can be tied in with another idea Say a set S is independent if s $dep S - {s} for all s in S Then the exchange axiom implies

24 General Fundamentals

(iv’) If x#dePS and S is independent then S u { x } is independent

(Indeed, suppose S u {x} is dependent Then some s Edep (S - {s}) u {x} and

by hypothesis s #dep S - {s}, so exchanging s and x yields x Edep (S - {s}) u

Conditions (i) and (ii) are usually immediate in any candidate for being a strong dependence relation However, (iii) and (iv) are trickier and may even

fail For example, one customarily says a subset S of a module M is indepen-

dent if xfinite rss = 0 necessarily implies each rs = 0 This definition satisfies (i), (ii) and (iv) but not necessarily (iii) On the other hand, if we instead try to

define x Edcp S by x E RS, then (iii) holds but (iv) may fail, e.g., R = M = Z,

x = 2, y = 4 and S = 0.) Fortunately, these two definitions coincide when R

is a division ring, so we get both (iii) and (iv)

Another example of interest to us is {simple submodules of a given R- module M} Then definition 1.3.5 below satisfies (i), (ii), and (iv), cf., argu- ment of remark 1.3.8 below, and (iii) holds because N n 2 Ni # 0 implies

N n

{s} = s

N, = N (since N is simple) and thus N c N,

Theorem 0.3.2: Suppose 9’ is a set with a strong dependence relation Edep

Then any independent subset of 9’ can be expanded to a suitable maximal

independent subset B, and s Edep B for every s E Moreover, if B‘ is another maximal independent subset then IBI = IB‘J

Proof: The existence of maximal independent subsets follows from Zorn’s lemma and property (ii), and if s & p B then B u {s} is independent by (iv’), contrary to the maximality of B Finally, we prove IBI 5 IB’I in two stages First assume B is finite, i.e., B = {b, , , b,} Then we claim there

are (b’, , , bb} in B’ such that {b’, , , b;,bk+, , , b,} is independent, for

each k I n Taking k = n this will prove {b’,, .,bb} is independent, so

IB’I 2 n = (BI

The claim is proved by induction on k For k = 0 this is clear, so suppose

we have found b’, , , &-, with {b; , , b,-,,b, , , b,} maximal inde- pendent; we look for bl Well b, Edep B’ and b, #dep {b’,, , b:- I , b,, ,, , b,},

so by (iii) we conclude there is some b‘ in B‘ with b’ &ep {b’,, ., bl- ,,

b,, ,, , b,}, and we take b’, to be this b’ {b;, ,4, b,, ,, , b,} is inde-

pendent by (iv’), as desired

Next assume B is infinite Write B‘ = { b’: i E I ’ } For each i in I’ take a finite

subset B, of B such that b: edep B, Then bl edep uisl, B, for each b’ in B‘

Since b Edep B’ for each b in B, we have b Edep u Bi for each b in B,

proving Uie,, B, = B Hence there are an infinite number of B,, proving

50.3 Abstract Dependence 25

IB’I is infinite, so then letting xo denote the cardinality of N we have

IB’I = /B’Ixo 2 xis,, (Bil = I u Bil = IBI

We have proved IBI I (B’I in all cases; by symmetry IB’I 5 IBI, so (BI = IB‘I Q.E.D

Remark 0.3.3: P M Cohn has observed that the conclusions of theo- rem 0.3.2 still are valid if we weaken (iii) by stipulating in its hypothesis that S

is also independent Let us call this a weak dependence relation

Algebraic and Transcendental Elements

One of the standard applications of the theory of abstract dependence is the study of transcendence bases in field theory Since we shall rely on this theory when examining extensions of rings (first in 42.5 and more extensively in Chapter 7), let us review the fundamentals from the commutative theory Let

C dtf denote the category of algebras over a commutative ring C

Given R in C - d t g and r E R we write C [ r ] for ( c : = , c i r i : c i E C , t E N)

C [ r ] is a commutative subalgebra of R, and there is a surjection q,:

C [ I ] t C [ r ] given by qrl = r, where I is a commuting indeterminate over

C; the elements of ker qr are the polynomials satisjed by r We say r is trans- cendental over C if ker qr = 0; otherwise r is algebraic over C, and we say r is integral over C iff r satisfies a monic polynomial R is integral (resp algebraic) ouer C if each element of R is integral (resp algebraic) over C When C is a field the notions, “algebraic” and “integral” coincide

More generally suppose the elements rl, , r, of R commute with each other Let CIIl, , I , ] denote the algebra of polynomials in the commuting indeterminates Al, , I , over C Writing C[rl, ,r,] for the (commutative)

C-subalgebra of R generated by rl, , rt we have the canonical surjection

q: C [ I , , , I , ] + C[r,, ,r,] given by q I i = ri for 1 5 i I t; we say r l , , r,

are algebraically independent (over C ) if ker q = 0

Example 0.3.4: We are ready to apply theorem 0.3.2 to the following situation: C c H are integral domains, whose respective fields of fractions are denoted F and K Algebraic dependence over F (in K) is a strong dependence relation, so any set of algebraically independent elements of K can be expanded to a maximal independent set, which is called a transcendence base

of K over F Moreover, the cardinality of a transcendence base of K over F

is unique and is called the transcendence degree of K over F, or tr deg KIF

Note that elements of H are algebraically independent over C iff they are

26 General Fundamentals

algebraically independent over F, seen by “clearing denominators,” so we can

also define tr deg HIC to be tr deg KIF

At times it is convenient to have the following generalization of “separable” for field extensions which are not necessarily algebraic; We say K 3 F is

separable if either char(F) = 0 or char(F) = 0 and K is linear disjoint from

F’Ip, the field obtained by adjoining p-th roots of all elements of F In other words if al, ,a, E K are linearly independent over F then a:, , a,P are also linearly independent over F It follows that every subfield K O of K finitely

generated over F is separably generated in the sense that K O is separably

algebraic over a purely transcendental field extension of F The standard proof, which can be found in Lang (65B, p.265) or Jacobson (80B, p.520) is an induction on the number of generators of K over F

In particular, any extension field of a perfect field is separable We shall see

more general uses of the word “separable” in 52.5 and then 55.3

Exercises

w.0

1 Given a set S with a preorder <, define an equivalence on S by stipulating a = b

iff a < b and b I a Then the set of equivalence classes is a poset, where [a] I [b]

iff a I b This observation is useful in generalizing results about posets

1’ Define < on Z by a s b iff b divides a Use this to define a PO on N via exercise 1, and identify this poset with Y(Z)

2 A lattice Y is modular iff Y has the property: If a 5 b and a A c = b A c and

a v c = b v c then a = b (Hint: (-=) Let a , = a v (b A c) and a, = b A (a v c)

Then a, I a, by remark 0.0.11 To show a2 _< a, one needs only show a, A c 2

a, A c and a, v c 2 a, v c.)

3 We say b covers a if a < b and there is no x with a < x < b Using exercise 2 show

for any a, b in the modular lattice Y there is a lattice isomorphism from the sub- lattice {x E 9 a < x < a v b to the sublattice {x E 4p:a A b < x < b}, given by

x -P x A b, with inverse function y + a v y (Hint: these are order-preserving) Conclude that if a v b covers a then b covers a A b

Many module theoretic results can be proved using lattice theory alone In the next few exercises we sketch the development of this aspect of lattice theory, foreshadowing results to appear later in the text (esp in Chapter 2)

4 A complete lattice Y is upper continuous if for every directed subset S of Y we

have a A (v S ) = v A s) for each a in 9, lower continuous is defined dually Show the modular lattice Y ( M ) is upper continuous but nor lower continuous

5 An element a in a complete lattice Y is compact if for each directed set {b,: i E I} with v b, 2 a we have some b, 2 a; Y is compactly generated if every element is

a supremum of compact elements Show every (complete) compactly generated lattice is upper continuous On the other hand, the compact elements of Y ( M )

are the f.g submodules of M, so Y ( M ) is compactly generated

Exercises 27 Modular Lattices In exercises 6 through 13 assume 9 is a modular lattice

6 The dual of Y is modular Every interval of Y is a modular lattice, which is complemented if U is complemented

7 Complements need not be unique in a complemented modular lattice (Take sP(R‘z’).) However, if a < b in 9 then no element can be a complement for both

a and b

8 If (a v b) A c = 0 then a A (b v c) = a A b

(Hint: a A (b v c) I (a v b) A (b v c) = b v ((a v b) A c) = b.)

9 An element a of U is large (also called essential) if a A b # 0 for all b # 0 If b is

maximal such that a A b = 0 then a v b is large (Hint: use exercise 8.)

10 If 9 is upper continuous then for any a, b in Y with a A b = 0 there is c 2 b in

Y such that a A c is large

An atom is a cover of 0 A lattice is atomic if for every a # 0 there is an atom

a, I a

11 Suppose Y is compactly generated 9 is complemented iff every element is a supremum of a suitable set of atoms (Hint: (*) Any compact element c has some

element b over which c is a cover, so c A b’ is an atom.)

12 The socle of 2, denoted soc(9), is defined as A{large elements of 9) The interval from 0 to soc(9) is a complemented lattice (Hint: Use exercise 9.) soc(9) 2 V{atoms of U } , equality holding if Y is compactly generated (by exercise 11)

13 Suppose Y is modular and complemented Y satisfies ACC iff Y satisfies DCC

(Hint: Given an infinite chain a , > a, > build an infinite ascending chain using

complements and exercises 6 thru 9 The reverse direction follows from duality.)

15 Prove the validity of transfinite induction (Hint: If S # a take a‘ minimal in

a - S and prove the absurdity a’ E S since fl E S for all /? < a’.)

16 If a # a’ are ordinals then either a < a’ or a’ < a (Hint: Take a minimal counter- example a’.)

17 Let B = {class of ordinals} 0 is not a set (for otherwise 0 E 8, which is impossible.) Thus to show a class GR is not a set it suffices to find a 1:l function from 0 to GR

18 Any set S is in 1 : 1 correspondence with a suitable ordinal (Hint: Otherwise define

a 1: 1 function f: B + S as follows: fb = so for some so in S and, by transfinite

induction, take fa to be some element of S - {ffl:fl E a } for each ordinal a.)

Conclude that every set can be well-ordered with respect to suitable total order!

19 For any set S, every function f: S + B ( S ) is not onto (Hint: {s E S : s $ fs} is not

in the image of f.) Consequently there is no 1: 1 function B ( S ) + S

20 Prove the maximal principle (Hint: If ( S , I) were an inductive poset without maximal elements take an ordinal a of the same cardinality as B ( S ) and define

j a + S by transfinite induction, taking fa‘ to be any element > sup{f/?: fl < a’}

for each ordinal a’ < a Then f is 1:1, contrary to exercise 19.)

28 General Fundamentals Distributive Lattices and Boolean Rings

21 A lattice 9 is distributive if a A ( b v c) = (a A b) v (a A c) for all a, b, c in Y Show

any distributive lattice satisfies the dual property a v ( b A c) = (a v b) A (a v c)

Y ( Z ) is a distributive lattice, but Y(,M) need not be distributive in general

22 A lattice is Boolean if it is distributive and complemented By a misnomer, Boolean lattices are usually called Boolean algebras; they are used in computer science

and circuitry Show any Boolean lattice Y has a ring structure given by ab = a A b

and a + b = (a A b’) v (a’ A b) (Hint: The hard axiom to verify is associa-

tivity of addition, which is proved by showing the function (a, b,c) + (a + b) + c

is symmetric in a, b, and c Indeed, a + b = (a v b) A (a’ v b‘) so (a + b) + c =

(a A b A c’) v (a’ A b A c’) v (a’ A b’ A c) v (a A b A c).)

23 An element e of a ring R is idempotent if e’ = e; we say R is a Boolean ring if every

element is idempotent Show every Boolean ring R is commutative, and r + r = 0

for all r in R (Hint: rl + r , = (rl + r$; then take r2 = 1.) Every Boolean lattice gives rise to a Boolean ring by exercise 22

24 For any ring {idempotents of Z ( R ) } form a Boolean lattice under the partial order

el < e, iEe,e, = e,

25 Suppose Y is a Boolean lattice Viewing Y as a ring as in exercise 22 show I Y

as ring iff I is a filter of the dual lattice of 9

26 The only Boolean integral domain is Z/2Z since e(l - e) = 0 But every homo- morphic image of a Boolean ringR is Boolean, so for A Q R it follows RIA is an integral domain iff RIA is a field

1 Monics in 9 i ~ q are all 1 : l (Hint: If f: R + R‘ is monk define a ring structure on the Cartesian product R x R by componentwise operations and let T = { ( r l r r 2 ) E

R x R: f r , = f r , } Then in1 = fn, where ni: T+ R is the projection on the i com-

ponent.) The proof of this exercise introduces two important constructions-the direct product and the pullback

2 Given an object A of V one can define the constant functor FA: I + V given by

F’i = A for all i E I and FA f = 1 , for all morphisms f Any morphism g : A + B

yields the natural transformation g: FA + Fs sending i to g Now define the

diagonal functor 6: V + given by 6A = FA and 6g = g

3 Interpret example 0.1.13 when I is a set with the discrete PO, viewed as a small category as in example 0.1.1

1 By comparing annihilators, show Z has nonisomorphic simple modules

2 If f : M + N is epic and kerf and N are f.g R-modules then M is also f.g

3 If M is a f.g R-module and A Q R is f.g as left ideal then AM is f.g (Hint: If M = 1 Rxi then AM = 1 Axi.)