tương đương morita cho nửa vành và đặc trưng một số lớp nửa vành bản tóm tắt luận án tiếng anh

In Chapter 2, we establish that ideal-simpleness, congruence-simpleness andsimpleness of semirings are Morita invariants for semirings; and describe allsimple semirings having projective

TRN GIANG NAM

MORITA EQUIVALENCE FOR SEMIRINGS ANDCHARACTERIZE SOME CLASSES OF SEMIRINGS

ABSTRACT OF THE DOCTORAL THESIS

Speciality: Algebra and theory of numbers

Code: 62.46.05.01

Supervisors

1 ASSOC PROF SCI DR NGUYEN XUAN TUYEN

2 ASSOC PROF DR NGO SY TUNG

VINH - 2011

Semiring is introduced by Vandiver in 1934, generalize the notion of commutative rings in the sense that negative elements don’t have exist In thisthesis, semirings were assumed to have both additive identity and multiplicativeidentiy

non-Nowadays one may clearly notice a growing interest in developing the braic theory of semirings and their numerous connections with, and applications

alge-in, different branches of mathematics, computer science, quantum physics, andmany other areas of science (see, for example, the recently published Glazek(2002)) As algebraic objects, semirings certainly are the most natural gen-eralization of such (at first glance different) algebraic systems as rings andbounded distributive lattices As is well known, structure theories for alge-bras of classes/varieties of algebras constitute an important “classical” area ofthe sustained interest in algebraic research In such theories, so-called simplealgebras — algebras possessing only the identity and universal congruences —play a very important role of “building blocks.” In contrast to the varieties ofgroups and rings, research on simple semirings has been just recently startedand, therefore, not much on the subject is known (for some recent results onthis subject one may consult Bashir et al (2001), Monico (2004), Bashir-Kepka(2007) and Zumbragel (2008))

In the same time in the modern homological theory of semimodules oversemirings, the results characterizing semirings by properties of semimodulesand/or suitable categories of semimodules over them are of great importance andsustained interest (for some recent such results one may consult, for example,Il’in and Katsov (2004), S N Il’in (2008), Y Katsov (2004), O Sokratova(2002))

At any rate, this thesis concerns the ideal- and congruence-simpleness- in asemiring setting, these two notions of simpleness are not the same and should bediffered-for some classes of semirings Also, comfirming a conjecture and solving

a problem of Katsov in the class of additively regular semisimple semirings.The thesis is organized as follows In Chaper 1, we present the number ofnew results on projective semimodules, which extend to a non-additive semiringsetting the corresponding classical important facts on projective modules over

rings We develop from two different, eventually equivalent, perpectives the

“Morita equibalence” concept in the category of semirings

In Chapter 2, we establish that ideal-simpleness, congruence-simpleness andsimpleness of semirings are Morita invariants for semirings; and describe allsimple semirings having projective minimal left (right) ideals, and provide acharacterization of ideal-simple semirings having projective minimal left (right)ideals

In Chapter 3, we describe all ideal-simple and simple artinian aic-semirings,

as well as congruence-simple lattice-orderd semirings Further, we also present

a complete desciription of left (right) subtractive semisimple semirings Then,apply this result, we describe left (right) subtractive artinian ideal-simple semir-ings, as well as left (right) subtractive artinian congruence-simple semirings.Finally, we describe all simple semirings with an infinite element, and provide acharacterization of ideal-simple semirings with an elements

In Chapter 4, we describe all additively idempotent semisimple semiringsover that the notions of either projectiveness and flatness, or flatness and mono-flatness for semimodules coincide Also, we characterize semisimple semirings

by projective and injective semimodules

From now on, let MRandRM be the categories of right and left semimodules,respectively, over a semiring R.

1.2 Progenerators

In this section, we characterize, and describe progenerators — finitelygenerated projective generators — of semimodule categories

Now let S = RM(P, P ) := End (RP ) be a semiring of all endomorphisms

of a left R-semimodule RP ∈ |RM| Then, considering endomorphisms of

RP operating on the right of RP , one easily sees that P becomes an semimodule, i.e., RPS ∈ |RMS| As in the classical case of modules over rings,

R-S-we write Q = P∗ := RM(RP,RR) for the dual S-R-semimodule P∗ of the S-semimodule P We also define the endomorphism qp ∈ S by p

R-0

(qp) = (p

0

q)pfor any p ∈ P and q ∈ Q Then, in the same fashion as has been done in theclassical case of modules over rings, one can easily obtain the following

Lemma 1.2.2 The assignments (p, q) 7−→ pq and (q, p) 7−→ qp definethe (R, R)-homomorphism α : P ⊗S Q −→ R and the (S, S)-homomorphism

β : Q ⊗RP −→ S, respectively.

Our next observation provides a characterization of finitely generated jective left R-semimodules RP ∈ |RM| by means of the homomorphism

pro-β : Q ⊗RP = P∗⊗RP −→ S = End (RP ) from Lemma 1.2.2 above

Proposition 1.2.4 A left R-semimodules RP ∈ |RM| is finitely generated andprojective iff β : Q ⊗RP −→ S is a surjection

Now we turn to generators in the semimodule categories

Definition 1.2.6 A left semimodule RP ∈ |RM| is said to be a generator forthe category of left semimodules RM if the regular semimodule RR ∈ |RM| is

a retract of a finite direct sum ⊕iP of the semimodule RP

Similar to Propositions 1.2.4, our next observation provides a characterization

of finitely generated generator RP ∈ |RM| for RM by means of the phism the (R, R)-homomorphism α : P ⊗SQ −→ R from Lemma 1.2.2 above

homomor-Proposition 1.2.8 A finitely generated left semimodule RP ∈ |RM| is agenerator for RM iff the (R, R)-homomorphism α : P ⊗S Q −→ R is asurjection Moreover, if α is a surjection, then it is an isomorphism

Combining the concepts of ‘generator’ and ‘finitely generated projectivesemimodule’, we come up with a new concept of ‘progenerator’, namely: Aleft semimodule RP ∈ |RM| is said to be a progenerator for the category of leftsemimodules RM if it is a finitely generated projective generator Then, fromPropositions 1.2.4 and 1.2.8, one obtains the following important result

Theorem 1.2.9 A left R-semimodules RP ∈ |RM| is a progenerator iff thehomomorphisms α : P ⊗SQ −→ R and β : Q ⊗RP −→ S are bisemimoduleisomorphisms

1.3 Morita Equivalence

The concept of Morita equivalence for semirings, or on the category SRing

of semirings, can be introduced in different fashions And for our goals we havefound that following approach to be good one

Definition 1.3.1 Semirings R and S is said to be Morita equivalent if thereexists a progenerator RP ∈ |RM| for RM such that the semirings S and End(RP ) are isomorphic, i.e., S ∼= End (RP ) as semirings

To present an alternative approach to the concept of Morita equivalence, weneed the following result that is is a nonadditive analog of the Eilenberg-Wattstheorem for module categories in the semimodule setting

Theorem 1.3.5 For a functor F : MR −→ MS the following statements areequivalent:

(i) F has a right adjoint;

(ii) F is right continuos and preserves coproducts (direct sums);

(iii) There exists unique up to natural isomorphism a R-S-bisemimodules

P ∈ |RMS| such that the functors − ⊗RP : MR −→ MS and F are naturallyisomorphic, i.e., F ∼= − ⊗RP

Using Theorem 1.3.5, one gets the following important and intriguing sequence of the Morita equivalence between two semirings R and S is the factthat reasonable important corresponding categories of semimodules over thesesemirings are equivalent, as categories, as well

con-Theorem 1.3.12 For semirings R and S the following conditions are lent:

equiva-(i) The semirings R and S are Morita equivalent semirings;

(ii) The semimodule categories MR and MS are equivalent categories;(iii) The semimodule categories RM and SM are equivalent categories

1.4 Conclusion

In this chapter, establishing a sufficient and necessary condition to a finitelygenerated semimodule is either projective (Propostion 1.2.4), or a generator(Proposition 1.2.8) Characterizing properties of progenerators of semimodulecategories (Theorem 1.2.9) Also, we develop from two different, eventuallyequivalent, perspectives the ‘Morita equivalence’ concept in the nonadditivesetting of the category of semirings Among several, as we hope useful forthe future research, observations obtained in this chapter, there are two centralresults — describing all covariant functors having right adjoints (Theorem 1.3.5),and characterizing Morita equivalence for semirings via semimodule categories(Theorem 4.12)

Congruence-simple and ideal-simple semirings have been studied by C ico (2004), J Zumbragel (2008), J Jeˇzek - T Kepka - M Mar´oti (2009), Bourne-Zassenhaus (1957), O Steinfeld - R Wiegandt (1967), Stone (1977), Weinert(1984),

Mon-A surjective homomorphism of semirings f : R −→ S is called stronglysemiisomorphic if Ker(f ) := f−1(0) = {0} and f (I) S for any proper ideal I

of R

A semiring R is called additively idempotent if the monoid (R, +) is additivelyidempotent Our next proposition illustrates that ideal-simple semirings can bestudied by additively idempotent simple semirings

Proposition 2.1.6 A semirng R is ideal-simple iff R is a simple ring, orthere exists a strong semiisomorphism from R onto an additively idempotent,simple semiring S

In the light of Proposition 2.1.6, it is natural to bring up that simple (simple) semirings can be understood by simpleness of subsemirings ofendomorphism semirings of idempotent commutative monoids Our next resultdescribes simple endomorphism semirings of idempotent commutative monoids.

ideal-Theorem 1.2.9 The following conditions for the endomorphism hemiringEnd(M ) of an idempotent commutative monoid (M, +, 0) are equivalent:

(i) End(M ) is simple;

(ii) End(M ) is ideal-simple;

(iii) The semilattice M is a finite distributive lattice

2.2 Morita invariants

Our next result establishes that ideal-simpleness, congruence-simpleness andsimpleness of semirings are Morita invariants for semirings

Theorem 2.2.6 Let R and S be semirings If R is Morita equivalent to S, then

R congruence-simple (ideal-simple) iff S is congruence-simple (ideal-simple); inparticular, R is simple iff S is simple

The following result shows that direct limits preverses ideal-simpleness,congruence-simpleness and simpleness of semirings

Proposition 2.2.7 Let {Ri| Ri ∈ |SR|, i ∈ I} be a directed family of semiringsand R = lim−→I Ri If Ri, i ∈ I, are (i) ideal-simple, (ii) congruence-simple, or(iii) left (right) subtractive, then R is ideal-simple, congruence-simple, or left(right) subtractive, respectively, too

Now let us consider an explicit, well known for rings, construction of the directlimit situation in a semiring setting Fix a semiring D and consider semirings

Ri = M2 i(D) (i ≥ 0) of square matrices of order 2i over D We shall regard Ri

as a subsemiring of Ri+1 by identifying a 2i× 2i matrix M with the 2i+1× 2i+1

matrix  M 0

 In this way, we have a chain of semirings

R0 ⊆ R1 ⊆ R2 ⊆ ,

where R0 = D; and it is clear that the direct limit R = lim−→I Ri of the directedfamily {Ri| i ∈ I} is, in fact, the union ∪I Ri of the semirings Ri, i ∈ I

Proposition 2.2.8 The semiring R = lim−→I Ri is not left artinian

In light of Propositions 2.2.7 and 2.2.8 and taking into consideration that evenfor rings the structure of general Congruence-simple, ideal-simple and simplerings remains difficult Therefore, in the next section and chapter, we onlystudy the above semirings for some special classes of semirings

2.3 Applications

Our following observation is an analog of the famous “Double CentralizerProperty” of ideals of simple rings in the setting of simple semirings

Theorem 2.3.1 Let R be a simple semiring, and I be a nonzero left ideal Let

D = End (RI) (viewed as a semiring of right operators on I) Then

(i) The natural map f : R −→ End (ID) is a semiring isomorphism;

(ii) I is a generator in the category of semimodule RM, and a finitelygenerated projective right D− semimodule;

(iii) There exists a natural number n and an idempotent e in matrix semiring

Mn(D) such that R ∼= eMn(D)e;

(iv) D is simple iff I is a finitely generated projective left R-semimodule

As a corollary of Theorem 2.3.1, we obtain the description of all simplesemirings having projective minimal left (right) ideals

Theorem 2.3.2 For a semiring R, the followings are equivalent:

(i) R is a simple semiring containing a projective minimal left ideal;

(ii) R is a simple semiring containing a projective minimal right ideal;

(iii) R is either isomorphic to a matrix semiring Mn(F ) for some divisionring F and n ≥ 1 , or ismorphic to EM, where M is a nonzero finite distributivelattice.

Combining Theorem 2.3.2 and Proposition 2.1.6, we give a description of anideal-simple semiring containing a projective minimal left (right) ideal

Theorem 2.3.4 Let R be a semiring containing a projective minimal left(right) ideal Then, R is ideal-simple if and only if it is isomorphic to the matrixring Mn(F ) for some division ring F, or there exists a strong semiisomorphismfrom R onto the endomorphism semiring End(M ) of a nonzero finite distributivelattice M

2.4 Conclusion

In this chapter, characterizing ideal-simple semirings via simple rings andadditively idempotent simple semirings (Prop 2.1.6), and describing all simpleendomorphism semirings of idempotent commutative monoids (Theorem 2.1.9).Also, establishing that ideal-simpleness, congruence-simpleness and simpleness

of semirings are Morita invariants for semirings (Theorem 2.2.6)

Describing all simple semirings having projective minimal left (right) ideals(Theorem 2.3.2), and provide a characterization of ideal-simple semirings havingprojective minimal left (right) ideals (Theorem 2.3.4)

Chapter 3

Simpleness for some classes of

semirings

3.1 Lattice-ordered semirings

Obviously, the additive reduct (R, +, 0) of an additively idempotent semiring

R in fact forms an upper semilattice and there exists the partial ordering ≤ on

R defined by for any x, y ∈ R, x ≤ y iff x + y = y If any two elements x, y ∈ R

of the poset (R, ≤) are comparable, the partial order relation ≤ is said to betotal, (R, ≤) forms a chain, and the semiring R is called an additively idempotentchain semiring or, in short, aic-semiring, by Takahashi-Wang (1993)

Let G be a totally-ordered multiplicative group and R := G ∪ {0} Then,extending the order on G to R by setting 0 ≤ g for any g ∈ G, and defining0g = g0 = 0 for all g ∈ G, one has that (R, max, ·) is a division aic-semiring,and called ”max–plus” semiring Kt qu di y m t cu trc ca na vnh Artin tri (phi)xch khng c ian khng tm thng v na vnh Artin tri (phi) xch n

Our following result describes all ideal-simple and simple artinian semirings

aic-Theorem 3.1.4 (i) A left (right) artinian aic-semiring R is ideal-simple iff it

is a division aic-semiring

(ii) A left (right) artinian aic-semiring R is simple iff R ∼= B

A semiring R is lattice-ordered if and only if there also exists the lattice