W e have to show th at the algebra -4 is S-regular... Since radical cla sses are closed under extensions, th e algebra A is 5-regular.. By th e second isom orphism theorem we have... The
Trang 1VNU JOURNAL OF SCIENCE, Mathematics - Physics, t XVIII n°l - 2002
ON T H E RA D IC A L C H A R A C T E R IS T IC OF R E G U L A R IT IE S
T r a n T r o n g H u e
Faculty o f M athem atics College o f N atural Sciences, V ietnam National University Hanoi
I I n t r o d u c t i o n
In th is paper we shall work in the variety w o f algebra s over an associative and com m utative ring A which unity elem ent For a given subclass I Ỉ o f th e variety 11 <‘ach algebra /1 6 R is called Ỉ Ì - algebra, and an ideal I o f an algebra A is called R r ideal if 1
is an //-algeb ra.
Radical classes are meant ill the sense o f K urosh [11] and A m itsur [1] and for d etails
of radical theory we refer to '18] and [19] It is well known th a t a n o n -em p ty subclass R
is a radical class in w if and only if the following con d ition s are satisfied.
(i) R is hornornorphically closed.
(ii) T h e sum R ( A ) o f all /ỉ-id e a ls o f an algebra A is an R rideal.
(iii) l i is closed under exten sions, th at is, if both I and A / I are i?- algebra s, th en ,4 is also /?- algebra.
In ring th eory much so - called regularity appear T h e oldest one seem s to be th e von Neum ann regularity In 1936 Von N eum ann [14] defined a lin g A (w ith identity) to
he regular if and only if for any elem ent a o f A there e x ists an elem en t X o f -4 such that
a z= (IX(I In 1950 Brown and M ccoy [6] generalized to rings w ith o u t identity, and th ey siiceeerU'il in proving that it is a radical property In the m eantim e (1942) Perils [ 15] had
in tro d u m l tli(' concept of quasi-regularity for algebra w ith identity He defined an algebra
A with identity to be quasi-regular if and only if for any elem ent X of' A there exist an elem ent ỊỊ o f A such th at x + y + x y = 0 In 1945 Jacobson [10] generalized th is concept t o arbitrary ring w ithout identity, and he show ed that it is a radical property, called later oil the Jacobson radical In 1948 Brown and M ccoy [5j a ttem p ted for define a general concept
of regularity for rings T h a t is the Brow n-M cC oy radical At th a t tim e their theory was gene nil enough All regularities introduced up to then were regularities ill th e sense o f it However, in 1971 a w ide class o f regularities was introduced by C ollid in g and Ortiz [8]
M cKnight and M usser [12], M usser [13] nam ely th e so-called ( p ,q) - regularities T h ey i;ul showed th at th e (p, q) - regularities are radical properties In 1975 R oos [16] gave
a general definition o f regularity for rings is term inology o f the class o f m appings { / r_4} - whore F a uiapps each ring A into th e set o f all subgroups o f the a d d itiv e group (A t) o f the ring A. T h e regularity in th is sense satisfies th e con d itions o f radical property In
1981 Hue and Szasa [91 gave th e definition of regularity o f a ssociative rings ill th e com m on term inology o f polynom ials and formal power series and showed th e radical characteristic
o f regularities in th is sense.
T he aim o f this paper is to give th e general d efin ition o f regularity for arbitrary algebra, which includes all regularities known up till now and to show th e radical charac teristic of regularities in th is sense We hope th a t we are going to get a diagram to define correct radical classes o f w
Typeset by -4Ạ/fS-rIteX 10
Trang 2O n the radical characteristic o f regularities
I I s - r e g u l a r i t i e s
1 D e f i n i t i o n s
s - n.ifulantf/
L<‘t then? !>(' assigned to ('ncii an algebra 4 belonging to w a mapping s \ which mapps the direct sum \ ' ~ -4r|./l,j into the algebra A. The class s consisting of all
m appings s,\ will 1><' milt'd a regularity for algebra o f \ v if the following condition is Silt isfi<‘(l.
Fur every /1 I Ì h ]\ and / £ Honi/v'(A ) D ). we have th e com m u tative diagram
,4 X 5a- > a
« x < -B
S i t
whore = ( / / ).
s - regular alợrhra
An elem ent a o f the algebra 4 is called 5-regular if a G 0 - s , \ An algebra 4 is said to 1)0 S -regulai if ẦS \ = A. All ideal / of an algebra A is called 5 - regular if I is an 5-rogular algebra.
2 P r o p e r t i e s
We obviou sly have the following.
P r o p o s i t i o n 1 I f tin class s IS a regularity then S /t ( ( 0 0 , ) ) = 0 f o r every algebra A
of the class IF.
P r o p o s it i o n 2 The class of the S-re.qular algebras is hom om orphically closed.
Proof. Let B be ail im age o f an 5-regular algebra A under the hom om orphism / Now let b be an arbitrary elem en t o f B T hen there ex ists an elem ent a o f A such
th at l> = / ( a ) S in ce A is an 5-regu lar algebra there is an elem ent X o f A such th at
s A ( x ) ~ a. B y th e com m u tative diagram (a) we have:
h = f(a) = f ( S A(;/•)) = S Bl f * ( x ) ) * S ) - B -
Therefore' th e algebra B is S-rogular.
T h e o r e m 3 I f the class s — { s \ : A ^ —> w4}.tei\ is a regularity then the class of all S-rcỊỊỉUar algebra IS a radical class in II’ i f and only i f the following condition is satisfied.
If I is an S -rcyular ideal of the algebra A and f o r every element a of A there exists
an (devient V o f A * such that S a (j:) - a = 0 m od I , then A is a regular algebra.
Proof. A ssu m e that th e class R o f all 5-regular algebras is a radical class Now supp ose that / is an 5-regular ideal o f an algebra and for every a o f A there ex ists an elem ent X of A'*' such th at S a ( x ) — a = 0 m od / W e have to show th at the algebra -4 is S-regular Let us consider th e fact or algebra A / 1 Take any elem en t ã o f A ị I By hypothesis there ex ists all elem ent X o f A00 such th a t S - A { x ) - a = 0 mod / So in
Trang 3\ 2 Tran 'Trong H u e
the factor algebra A / I th e equality (I = S a ( x ) holds For the natural hom om orphism
/) : /1 * A / I we have the com m u tative diagram
We have (1 = S a ( x ) = p(S.4(x )) = 5 ^ // ( p ° ° ( x ) ) Therefore th e elem en t Ỡ is S - regular T his im plies the 5-regularit.v o f th e algebra A / I Since radical cla sses are closed under extensions, th e algebra A is 5-regular.
Conversely, assum e th a t the 5-regu larity satisfies the condition o f th e theorem We shall show that the class R o f all 5-regular algebras is a radical class C learly, the class R
is not em pty.
By proposition 2 th e class R is honiom orphically closed T h e co n d itio n (i) o f th e radical property is satisfied.
Now su pp ose that for an ideal o f an algebra A, b oth / and A / I are /?-algebra Since the algebra A / I is 9-regular therefore for every elem ent a o f A th ere ex ists an elem ent 7 o f (A /I)'* - such th at S a / i {x ) = Ã By th e com m u tative diagram (b) we have
a = ^ / / ( x ) = S A/ i ( p° ° { x) ) = p(S>i(x)) = Srf(x) This implies s.,t(;r) - a s 0 mod /.
By the condition o f theorem th e algebra /1 is 5-regular H ence tho class R is closed under extensions T h e condition (iii) o f th e radical property is satisfied.
By th e proposit ion 1 th e zero ideal o f an algebra A is an 5-regu lar ideal Hence th e set of the R-ideals o f algebra A is not em pty S u p p ose both I\ and /*2 b e R-id eals o f th e algebra ,4 By th e second isom orphism theorem we have.
Since the class R is hom om orphicallv closed and closed under ex ten sio n s, th e above iso morphism im plies th a t /1 + 1 2 is an /ĩ-algeb ra By a sim ple induction we can prove that the sum o f any finite num ber o f I Ỉ -ideals o f th e algebra A is again an /?-ideal Filially,
we have to show that the sum R ( A ) o f ail fî-id eals o f th e algebra A is an 5-regular ideal Take any elem ent a o f R (A ) T hen there are thí' R n-ideals / 1 such th a t th e ideal
J = I h- contains the elem ent a. Sine J is an S-regular ideal th ere is an elem ent JT
of J x such th a t s / ( x ) = a For th e em b edd ing i j : J —> /? (v4 ) we have th e com m u tative diagram.
A 00
A / I
p
(b )
h ± h /
h 11 n /2
O f
Therefor th e ideal /Ỉ(A ) is 5-regular T h is co m p letes th e proof o f th e theorem
As th e radical criterions o f 5-regu larities we have th e follow ing assertions.
Trang 4ơ n the radical chavuctf'.ristic o f reg u la rities
P r o p o s i t i o n 4 The class of all S -rcỊỊu lar (dọcbras is a radical, class if the following con dition is satisfied.
F o r arbitrary elements (Ï ill an alqebra A, and X of A™ i f the element S \(:v) - a is
s ' I f (Ịiỉlíỉ I then iln ỊÉlcmrtì1 a is also s -vcọulav.
Proof. \Y<* are going to si low that th e con dition o f theorem 3 is valid Let 1 1><* call V-n‘pillar ideal o f an algebra A w ith the follow ing property: For CVCTV clem ent a o f 4
th ere e x ists an elem ent /• o f 1 v such th at s t(.r ) - tt = 0 m od I. Therefore1 the* elenirni
s \(.r) a b elongs to the ideal / S in ce / is an S-regular ideal so th e elem ent S,\(J') ft
is S-n'gular By hyp oth esis rlie elem ent a is S-ivgular Thus the» algebra 4 is S’-ro&ular
T h v co n d itio n o f T hooivni *{ is satisfied T h e proposition is proved.
P r o p o s i t i o n 5 77/f class of all s -re g u la r algebin.s is (I radical (‘lass if the followiiuj
c o n d i t i o n class i f thỉ foil o w n IỈỊ c o n d i t i o n is satisfied.
L i t I be can S - r ạ Ịu la r u h a l o f an a lijtb ra A I f the element a o f t.h.c factor alqcbru
A l is S -ra ju la r then tlỉỉè (l( lit( 711 a is s -re g u la r in the algebra A.
Proof. A ssum e that I is nil S'-regular ideal o f all algebra A. and for every (’loment
a of /1 th en ' e x ists nil elem ent r o f A'*' such th at s \(:r ) - a = 0 m od / Hence ill
I lie factor algobra A / I we have* s ,\{.r ) = a. B y th e com m u tative diagram (b) we have*
Ti = S a ( j -) = p { S A ( r ) ) = 5 i / , ( / / v (.r)).
T herefore ã is an 5-regular clem ent o f th e algebra A / I By hyp oth esis the elem ent (I
is 5-regu lar ill th e algebra /1 T hus th e algebra A is S-regular T he con dition o f Theorem
3 is valid T he proposition is proved.
I I I E x a m p l e s o f .S - r e g u la r itie s
111 this section wr shall list the ^ r e g u la r itie s which arc' known to us O ne had proved thạ th rse regu larities are the radical properties in the srnsc of Kurosli <111(1 A m itsur i.e
th ese w ell-know n regularities are th e S -regu larities satisfyin g th e con d ition o f Theorem
3 S u p p ose that u 0 is the variety o f a sso cia tiv e algebra For arbitrary su b sets A' and Y
o f an algebra A we denote -YV = { 5^;I=1 x *y* : x * £ X , y, € Y } Lot us consider som e follow ing regularities s k = —> A}.4çvr0 *
1 .(12 — )) =
A ll elem ent a o f an algebra A is said to be regular in the souse o f Neum ann [14| if
a € (lAa. Clearly 5 1-regular coincides w ith the regularity ill th e sense o f N eum ann.
2 6^ ((< 7i,tt2, ) ) = - («-I
T h e right quasi-regularity had been defined by Piỵlis [15] and later stud ied by Baer
|3] and Jacobson [10] All elem ent a o f ail algebra A is said to be right quasi-regular if
</ -f I) 4- ab = 0 for som e elem ent I) o f A. H ence s 2-regularity is right - qufusi regularity.
3 S :\((a\*U - 2 * ) ) = a -2 + = ^ x’a2/+ l a2(i+ l) + a2i-f l ar/,2(i + n
Brow n and M ccoy [5| have introd uced th e notion o f G -regularity An elem ent a of
an algebra A is said to 1)0 G -regular if th e elem en t a is ill G (a), where
G { a ) = /1(1 + u ) + A ( l -f a ) A
I t is c le a r t o S(»(' I lia i s,:ỉ- r ( 'g u l a r i t v is G - r e g u la r it y
Trang 54 s \ ( ( a|.a*2, )) = d]<v>
T he notion o f strongly regular algebra had been introduced by A rm s and K aplaiiskv 2] and was later studied by others An algebra A is strongly regular if a 6 a'2A for every
a £ A. It is clear that ^ -reg u la rity is the sam e as strong regularity.
D<* La Rose [17 has introduced th e notion o f À-ivgularitv An (‘l(*ni(‘iit (I o f an algebra 1 is A-regular if a € An A. Clearly 6v>-regularity is A-regularity.
6 S j {(a I a■>■■■■)) = -U -Ỉ(a 1 + à ị)
D ivinsky [7j has introd uced left pseudo-regularity An elem ent a o f an algebra A
is left pseudo-regular if a -f ba 4- her — 0 for som e clem ent I) o f A. It is easy to see that
^ ‘-regularity coincides w ith left pseudo-regularity.
7 s \ ( ( a J n-2 ) ) = z r I 1a 1 aMa 1 a'M+ 1
Blair 4| introduced the notion o f /-regu larity, which was later stu d ied by others All elem ent a o f 'Mị algebra A is said to be /-r e g u la r if a Ç (o'2), w here (a ) denotes the principal ideal o f A generated by a Blair has show n th at an clem ent a ill an algebra A is / - regular if and only if there exist elem en ts u - i, r, and in, in 4 such as a = X / - Ị u,aVi<iWj
Hence 5 ' -regularity is th e sam e as /-r e g u la r ity in the sense of Blair.
8 s 7'\((n [.<!■>< )) — p{<* w here p ( x ) and q {x ) are in the polynom ial ring A'[./:].
T he (/M /)-iv g u lw ity was introduced by M cknight and Mussel* [12] An algebra A
is (/>.</) - regular if the inclusion a € p (a )A q (a ) holds for every elem ent a of A. where
p(.r).q(.r) are ill /\ [./•] It is easy to see th a t s^ -regularity is (p q )- regularity ill the sense
of Kckilight and Musser.
T h e open problem s
P r o b l e m 1 F in d a necessary and sufficient condition f o r the s-re g u la rity is hereditary
P r o b l e m 2 Establish some diagrams to define concret radical classes by s-regularities.
R e f e r e n c e s
1 S.A A m itsur A general th eory o f radicals, i Amer J Math., 7 4 (1 9 5 2 ) 774-786.
2 R F Arons and I K ạplansky T opological representation o f algebras Trans Airier Math Sot' 6 3 (1 9 4 8 ), 457-481.
3 R Baer R adical ideals, Amur J Math 6 5 (1 9 4 3 ) 537 - 568.
I R.L Blair A note on f-regularity in rings Proc A m cr Math S oc 6 (1955) 511
r> B.Brow n and N.H M ccov T h e radical o f a ring D uke Math. 1 5 (1 9 4 8 ) 495 - 499.
G B Brown and N.H Mccoy T h e m axim al regular ideal o f a ring, Proc Amer Math Soc 1(1950), 165 - 171.
7 N D ivinsky Pseudo-regularity, Canad J Math., 7 (1958), 401 - 410.
Trang 6Oil the radical characteristic o f regularities r
N I L Colliding ami A II Ortiz Structure of st'tnipriim’ (]) q) - radicals I ’at. /
Math 3 7 (1 9 7 1 ), 97 - 9!).
9 T T Mue and F S /a s / Oil th e radical classes (k'tmniiiLHl by regularities Acta Sri M ad) 4 3 ( 1!)M ) i : h - I TO.
10 N .Jacobson Some rem arks on one-sided inverses, Pvoc Amur Math Sot'
1 (1 9 5 0 ) 352 - :r>:>.
11 A (Ỉ Kurosli R adicals o f rings and algebras Mat SI)., 3 3 (7 5 ) (1953) 13 - 20.
12 .) D Kcknight and G L M usser Special (p.q) - radicals Canad. •/ Math
2 4 (1 9 7 2 ) 3 8 - 44.
i:i c L M ussel- Linear scm iprim e (p q) -radical Pac. / Math 3 7 (1 9 7 1 ) 749 - 757.
1 I Von N eum ann On regular rings Proc Math Acad Sri 2 2 (1 9 3 0 ) 7 6 7 - 783.
15 S Perl is A charactrm aU on o f th e radical o f an algebra Bull Am er Math Sor.,
48(11)42) 128 - 132.
l(i c R ous lieyuluritic.s o f rings. D issertation D elft 1975.
17 13 D(' La R ose U rals and radicals. D issertation Delft 1970.
Is F A Szasz Radiacals of lings. Akadm iai Kind Budapest 1981.
19 H W’i«'•garnit Radical and sem isin ip le classes o f rings Quern's Papers III Pirn (111(1
A /ipl M oth V o l 3 7 ( K i n g s t o n 1 9 7 4 ).
TAP C.HI KHOA HOC DHQGHN Toán • Ly t XVIII n°l - 2002
VỀ Đ Ặ C TRUNc; C A N C Ủ A C Á C TÍN H CHAT C H ÍN H Q U Y
Trần Trong Huệ
Kltoa Toán - C ơ - Till học
D ili học K h o a học T ự Iiliiên - D H Q C Ì H ù N ộ i
XÓI \ v hì đa lạp các dại sỏ (k h ôn g nhất thiết kêì hợp) trẻn vành K giao hoán có
đ ơ n v ị V ớ i m ỗ i đ ạ i s ỏ A th u ộ c i r k ý h iệ u A * là t ổ n g tr ự c t iế p A „ p l u.sA.ij M ỗ i lớ p
sau dược thoá mãn:
Đùi với mọi A , B thuộc IV và / thuộc h o n i K ( A R ) ta có hộ thức giao hoán
f s ( - $ i t f x trong đó / x = ( / / .)• Phần tứ n cùa đại SC) 1 gọi là S- chính quy nêu (I ị- Ò.S' ( Đại s ố A aọi là 5 -c h ín h quy nêu 'iS\\ — A. Iđêiiii Ị cua đại s ố A gọi lit S-
cliính quy nếu 1 là một đại số S- chính quy.
Trong bài báo này chúng tỏi đã chứng minh được ràng tính chất s - chính quy là
I11ỘI l í n h c h á t c á n t h e o n g h ĩ a K u r o s h v à A m i t s u r k h i v à c h i k h i đ i ổ u k i ệ n s a u đ ư ợ c t h o i i
m ã n :
Nếu / là một iđêan s - chính quy của đại sô A và đối với mọi a € A tổn tại phần
tử /• f sao cho s.\(.r) - a = 0 mod I thì A là một đại số S - chính quy.
Từ đặc trưng này ta chứng minh dược hai điéu kiện đủ đổ một tính chất s - chính
q u y l à t í n h c h ấ t c ă n
Trong trườn” hợp 11' là đa tạp các đại sổ kết hợp thì khái niệm S - chính quy và
c á c k é t q u à c ú a b à i b á o n à y l à s ự t ổ n g q u á t h o á c á c t í n h c h á t c h í n h q u y c ù a c á c t á c g i á
Von N eum ann, Perlis (cán Jacobson) Brown - M cCoy, Kaplansky, D e la R ose, D ivinsky, Blair M aknish - M usser, v.v