HAZAR ABU-KHUZAM, HOWARD E. BELL, AND ADIL YAQUB
Received 24 February 2005
A ring is called semi-weakly periodic if each element which is not in the center or the Jacobson radical can be written as the sum of a potent element and a nilpotent element. After discussing some basic properties of such rings, we investigate their commutativity behavior.
1. Introduction
An elementxof the ringRis calledperiodicif there exist distinct positive integersm,n such thatxm=xn; andxispotentif there existsn >1 for whichxn=x. We denote the set of potent elements byPorP(R), the set of nilpotent elements byNorN(R), the center byZorZ(R), and the Jacobson radical byJorJ(R).
The ringRis calledperiodicif each of its elements is periodic, andRis calledweakly periodicifR=P+N. It is easy to show that every periodic ring is weakly periodic, but whether the converse holds is apparently not known. It has long been known that peri-odic rings have nice commutativity behavior; in particular, Herstein [10] showed that if Ris periodic andN⊆Z, thenRis commutative—a result which extends easily to weakly periodic rings. Various generalized periodic and weakly periodic rings have been intro-duced in recent years, and their commutativity behavior has been explored [6,7, 13,
14].
DefineRto besemi-weakly periodicifR\(J∪Z)⊆P+N. Clearly the class of semi-weakly periodic rings is quite large; it contains all semi-weakly periodic rings, all commutative rings, and all Jacobson radical rings. Our purpose is to point out some general properties of semi-weakly periodic rings and to investigate commutativity of such rings.
2. Preliminaries
We fix some more notation. Ifx,y∈R, the symbol [x,y] denotes the commutatorxy− yx; and ifS,T⊆R, [S,T] denotes the set{[s,t]/s∈S,t∈T}. Forx∈R, the symbolx denotes the subring generated byx; and the symbolsC(R) andEstand for the commu-tator ideal and the set of idempotents ofR. If℘is a ring property, a ringRhaving the property is called a℘-ring.
Copyright©2005 Hindawi Publishing Corporation
We also state some known results we require, the first of which is trivial.
Lemma2.1. IfRis any ring andSis any proper additive subgroup ofR, the centralizer of R\Sis equal toZ(R).
Lemma2.2 [9]. IfRis a ring such that for eachx∈R, there exists an integern >1such that xn−x∈Z, thenRis commutative. In particular, ifR=P∪Z, thenRis commutative.
Lemma2.3. IfRhas an idealIsuch thatIandR/Iare both commutative, thenNis an ideal andC(R)⊆N.
Proof. SinceR/Iis commutative, [x,y]∈I for allx,y∈R; and sinceI is commutative, Rsatisfies the polynomial identity [[x,y], [z,w]]=0, which is not satisfied by the ring of 2×2 matrices over anyGF(p). The result now follows by [1, Theorem 1].
Lemma2.4 [8]. LetRbe a ring such that for eachx∈R, there exist a positive integermand apolynomialp(X)with integer coefficients for whichxm=xm+1p(x). ThenRis periodic.
Lemma2.5 [4, Theorem 2]. LetRbe an arbitrary ring, and letN∗= {x∈R/x2=0}. If N∗is commutative andNis multiplicatively closed, thenPN⊆N.
We conclude this section with a theorem stating some basic results on semi-weakly periodic rings.
Theorem2.6. LetRbe a semi-weakly periodic ring. (a) Every ideal ofRis semi-weakly periodic.
(b) Every epimorphic image ofRis semi-weakly periodic.
(c) IfNis an ideal, then for eachx∈R\(J∪Z), there existsn >1for whichx−xn∈N. (d)Ris weakly periodic if and only ifZis periodic andJis nil.
(e) IfN⊆J⊆Z, thenRis commutative.
Proof. (a) LetI be an ideal of R andx∈I\(J(I)∪Z(I)). Clearlyx /∈Z(R); and since J(I)=I∩J(R), x /∈J(R). Therefore,x=a+u, where u∈N and a∈P; and we may choosen >1 such thatun=0 andan=a. It follows thata=an=(x−u)n∈Iand hence u∈I. Consequently,Iis semi-weakly periodic.
(b) LetSbe a ring and letϕ:R→Sbe an epimorphism. Lety∈S\(J(S)∪Z(S)), and letx∈ϕ−1(y). Sinceϕ(J(R))⊆J(S), we havex /∈J(R)∪Z(R) and thereforex∈P(R) + N(R). It follows thaty∈P(S) +N(S), henceSis semi-weakly periodic.
(c) Letx∈R\(J∪Z); and writex=a+u, whereu∈N andan=a,n >1. Thenx− xn=a−an+u−wwherew∈N, hencex−xn∈N.
(d) Clearly, ifZis periodic andJis nil, thenRis weakly periodic. Conversely, suppose Ris weakly periodic. By the argument in the proof of (a),Jis weakly periodic; and since J contains no nonzero idempotents and hence no nonzero potent elements,J is nil. Let z∈Zand writez=a+uwithu∈Nandan=a,n >1. Then [a,u]=0 and hencez−zn is a sum of commuting nilpotent elements, so thatz−zn∈N. It follows fromLemma 2.4 thatZis periodic.
3. Commutativity results
It is proved in [2] that ifRis periodic andNis commutative, thenNis an ideal. Surpris-ingly, this result extends to semi-weakly periodic rings.
Theorem3.1. LetRbe a semi-weakly periodic ring withR =J. IfN is commutative, then Nis an ideal.
Proof. SinceNis commutative,Nis an additive subgroup and is closed under multipli-cation. ByLemma 2.5,PN⊆N; and it follows that (R\(J∪Z))N⊆N. SinceZN⊆N, we have (R\J)N⊆N. Now letu∈Nandy∈J, and letx∈R\J. Thenx+y∈R\J, and hence yu=(x+y)u−xu∈N. Therefore,JN⊆Nand henceRN⊆N.
Corollary3.2. IfRis a semi-weakly periodic ring in whichJ is commutative andN is commutative, thenNis an ideal andC(R)⊆N.
Proof. IfR=J, thenRis commutative and the conclusion is immediate. If R =J,N is an ideal byTheorem 3.1and henceN⊆J. Therefore, inR/Jevery element is either po-tent or central, so thatR/Jis commutative byLemma 2.2. Our result now follows from
Lemma 2.3.
Herstein’s theorem on commutativity of periodic rings withN⊆Zalso has an exten-sion to semi-weakly periodic rings.
Theorem3.3. LetRbe a semi-weakly periodic ring withR =J. IfN⊆Z, thenRis commu-tative.
Proof. SinceN⊆Z,N is an ideal and henceN⊆J. By Theorem 2.6(c), for each x∈ R\(J∪Z), there existsn >1 such thatxn−x∈N⊆Z; moreover, sinceex−exeandxe− exeare inN for allx∈Rand alle∈E, we can use a standard argument to show that E⊆Z.
ByTheorem 2.6(e), we need only show thatJ⊆Z. Assume first thatRhas 1, and sup-pose thatw∈J\Z. Since 1∈/ J, we see at once that 1−w /∈J∪Z. It follows that there exist at most one prime q such thatq(1−w)∈J and at most one prime q such that q(1−w)∈Z, hence there exists a primepsuch thatp(1−w)∈/ J∪Z. Thus, there exists n >1 such that (1−w)n−(1−w)∈N and (p(1−w))n−p(1−w)∈N; consequently, (pn−p)(1−w)∈N. Since 1−w is invertible, this yieldsk >1 such that (pn−p)kR= {0}; and this fact, together with the fact that (1−w)n−(1−w)∈N, shows thatwis finite and hencew is periodic. But the only periodic elements inJare nilpotent, so we have contradicted our hypothesis thatw∈J\Z. Thus,J⊆Zas required.
Now supposeRdoes not have 1. IfR=J∪Z, then we must haveR=Z, since a group cannot be the union of two proper subgroups; therefore we may suppose thatR =J∪Z, in which caseP = {0}, sinceN⊆J. Leta∈P\{0}withan=a,n >1. Thene=an−1is an idempotent, necessarily central; and byTheorem 2.6(a),eRis semi-weakly periodic with multiplicative identityeand nilpotent elements central. Hence, by the argument above, J(eR)⊆Z(eR). Ifu∈J(R), theneu∈eR∩J(R)=J(eR), hence [ea,eu]=0=e[a,u]= [a,u]. Thus, [J(R),P+N]=0=[J(R),R\(J(R)∪Z(R))]=[J(R),R\J(R)], soJ(R)⊆Z(R)
byLemma 2.1.
Theorem3.4. LetRbe a semi-weakly periodic ring withR =J. IfN is commutative and each element ofR\(J∪Z)is uniquely expressible as a sum of a potent element and a nilpotent element, thenRis commutative.
Proof. We begin by showing thatE⊆Z—a fact which will enable us to pass from the case ofRwith 1 to the general case by an argument similar to that used in the proof of
Theorem 3.3.
Supposee∈E\Z. Then if [e,x] =0, eitherex−exe =0 orxe−exe =0; and we assume ex−exe =0, in which case the nonzero idempotent f =e+ex−exeis not inJ∪Z. Then we have (e+ex−exe) + 0=e+ (ex−exe)—two representations off as a sum of a potent element and a nilpotent element. Therefore,ex−exe=0—a contradiction.
It does not seem necessary to write out the details of the case R without 1, so we assume henceforth thatRhas 1. In view ofTheorem 3.3, we need only show thatN⊆Z. Suppose thatw∈N\Z. The same argument used in the previous proof shows that (R, +) is a torsion group and there existsn >1 such that (1 +w)n−(1 +w)∈N; and it follows that1 +wis finite. Thus, 1 +wis a periodic invertible element, that is, a potent element. Now 1 +w /∈J∪Z and (1 +w) + 0=1 +w, where 1 and 1 +w are inP, and 0 andw are inN; therefore w=0, contradicting our assumption thatw /∈Z. Thus, N⊆Z as
required.
It appears from our proofs that the serious work of establishing commutativity of semi-weakly periodic rings is proving the result forRwith 1. This observation suggests the following general theorem.
Theorem3.5. Let℘be a ring property which is inherited by ideals and which implies that N is an ideal, and suppose that every semi-weakly periodic℘-ring with 1 is commutative. IfR is any semi-weakly periodic℘-ring in whichE⊆Z andJ is commutative, thenRis commutative.
Proof. IfR=J∪Z, then R=J or R=Z and henceR is commutative. Otherwise, if a∈P\{0}, letan=aandan−1=e. Theneis a central idempotent; andeRis commu-tative, since it is a semi-weakly periodic℘-ring with 1. Thus [ea,ex]=0 for allx∈R, that is,an−1[a,x]=0=[a,x] for allx∈R. Therefore,P⊆Zand in particular [P,J]=0. SinceN⊆J, [N,J]=0 and we conclude that [R\J,J]=0, so thatJ⊆Z byLemma 2.1.
Commutativity ofRnow follows fromTheorem 2.6(e).
Of course there are many applications of this theorem, since there are many conditions known to imply commutativity of rings with 1 but not of arbitrary rings. We conclude with one example.
Theorem3.6. Letn >1be a fixed positive integer and let Rbe ann(n−1)-torsion-free semi-weakly periodic ring withE⊆ZandJcommutative. If
(∗)
(xy)n=xnyn ∀x,y∈R\N, (3.1)
thenRis commutative.
(∗) is commutative. Hence, our theorem follows fromTheorem 3.5once we show that property℘forcesNto be an ideal. In fact, we show that (∗) together with the hypothesis thatJis commutative forcesNto be an ideal. ConsiderR/J, which is a subdirect product of primitive ringsRαsatisfying (∗). Since (∗) is inherited by subrings and epimorphic images, it follows from Jacobson’s density theorem that either allRα are division rings, or there exist a division ringDand an integerm >1 for which the ring ofm×m matri-ces overDsatisfies (∗). A simple substitution shows that the second alternative cannot occur, so eachRαis a division ring such that (xy)n=xnynfor allx,y∈Rα. But such di-vision rings are commutative by a well-known theorem of Herstein [11,12], soR/J is
commutative and henceNis an ideal byLemma 2.3.
Acknowledgment
Professor Bell’s research was supported by the Natural Sciences and Engineering Research Council of Canada, Grant no. 3961.
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Hazar Abu-Khuzam: Department of Mathematics, American University of Beirut, Beirut 1107 2020, Lebanon
E-mail address:hazar@aub.edu.lb
Howard E. Bell: Department of Mathematics, Brock University, St. Catharines, ON, Canada L2S 3A1
E-mail address:hbell@brocku.ca
Adil Yaqub: Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
