STRUCTURE OF RINGS WITH CERTAIN CONDITIONS ON ZERO DIVISORS

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STRUCTURE OF RINGS WITH CERTAIN CONDITIONS ON ZERO DIVISORS

HAZAR ABU-KHUZAM AND ADIL YAQUB

Received 4 May 2004; Revised 17 September 2004; Accepted 24 July 2006

LetRbe a ring such that every zero divisorxis expressible as a sum of a nilpotent element and a potent element ofR:x=a+b, whereais nilpotent,bis potent, andab=ba. We call such a ring aD^∗-ring. We give the structure of periodicD^∗-ring, weakly periodic D^∗-ring, ArtinianD^∗-ring, semiperfectD^∗-ring, and other classes ofD^∗-ring.

1. Introduction

Throughout this paper,Ris an associative ring; andN,C,C(R), andJdenote, respectively, the set of nilpotent elements, the center, the commutator ideal, and the Jacobson radical.

An elementxofRis called potent ifxⁿ=xfor some positive integern=n(x)>1. A ringR is called periodic if for everyxinR,x^m=xⁿfor some distinct positive integersm=m(x), n=n(x). A ringRis called weakly periodic if every element ofRis expressible as a sum of a nilpotent element and a potent element ofR:R=N+P, wherePis the set of potent elements ofR. A ringRsuch that every zero divisor is nilpotent is called aD-ring. The structure of certain classes ofD-rings was studied in [1]. Following [7],Ris called normal if all of its idempotents are inC. A ringRis called aD^∗-ring, if every zero divisorx in Rcan be written asx=a+b, wherea∈N,b∈P, andab=ba. Clearly everyD-ring is aD^∗-ring. In particular every nil ring is aD^∗-ring, and every domain is aD^∗-ring. A Boolean ring is aD^∗-ring but not aD-ring. Our objective is to study the structure of certain classes ofD^∗-ring.

2. Main results

We start by stating the following known lemmas: Lemmas2.1and2.2were proved in [5], Lemmas2.3and2.4were proved in [4].

Lemma 2.1. LetR be a weakly periodic ring. Then the Jacobson radical J ofR is nil. If, furthermore,xR⊆Nfor allx∈N, thenN=JandRis periodic.

Lemma 2.2. IfRis a weakly periodic division ring, thenRis a field.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 67692, Pages1–6

DOI 10.1155/IJMMS/2006/67692

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Lemma 2.3. LetRbe a periodic ring andxany element ofR. Then (a) some power ofxis idempotent;

(b) there exists an integern >1 such thatx−xⁿ∈N.

Lemma 2.4. LetRbe a periodic ring and letσ:R→Sbe a homomorphism ofRonto a ring S. Then the nilpotents ofScoincide withσ(N), whereNis the set of nilpotents ofR. Definition 2.5. A ring is said to be aD-ring if every zero divisor is nilpotent. A ringR is called aD^∗-ring if every zero divisorxinRcan be written asx=a+b, wherea∈N, b∈P, andab=ba.

Theorem 2.6. A ringRis aD^∗-ring if and only if every zero divisor ofRis periodic.

Proof. AssumeRis aD^∗-ring and letxbe any zero divisor. Then

x=a+b, a∈N,b∈P,ab=ba. (2.1) So, (x−a)=b=bⁿ=(x−a)ⁿ. This implies, sincexcommutes witha, that (x−a)= (x−a)ⁿ=xⁿ+ sum of pairwise commuting nilpotent elements.

Hence

x−xⁿ∈N for every zero divisorx. (2.2) Since each such x is included in a subring of zero divisors, which is periodic by Chacron’s theorem,xis periodic.

Suppose, conversely, that each zero divisor is periodic. Then by the proof of [4, Lemma

1],Ris aD^∗-ring.

Theorem 2.7. IfRis any normalD^∗-ring, then eitherRis periodic orRis aD-ring. More- over,aR⊆Nfor eacha∈N.

Proof. IfRis a normalD^∗-ring which is not aD-ring, thenRhas a central idempotent zero divisore. ThenR=eR⊕A(e), whereeRandA(e) both consist of zero divisors ofR, hence (in view ofTheorem 2.6) are periodic. ThereforeRis periodic.

Now considera∈N andx∈R. Sinceaxis a zero divisor, hence a periodic element, (ax)^j=e is a central idempotent for some j. Thus (ax)^j+1=(ax)^jax=a²y for some y∈R. Repeating this argument, one can show that for each positive integerk, there exists msuch that (ax)^m=a²^kwfor somew∈R. ThereforeaR⊆N.

Corollary 2.8. LetRbe aD^∗-ring which is not aD-ring. IfN⊆C, thenRis commutative.

Proof. SinceN⊆C,Ris normal. Therefore commutativity follows fromTheorem 2.7and

a theorem of Herstein.

Now, we prove the following result forD^∗-rings.

Theorem 2.9. LetRbe a normalD^∗-ring.

(i) IfRis weakly periodic, thenN is an ideal ofR,Ris periodic, andRis a subdirect sum of nil rings and/or local ringsRi. Furthermore, ifNiis the set of nilpotents of the local ringRi, thenRi/Niis a periodic field.

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(ii) IfR is Artinian, thenN is an ideal andR/N is a finite direct product of division rings.

Proof. (i) UsingTheorem 2.7, we have

aR⊆N for everya∈N. (2.3)

This implies, usingLemma 2.1, thatN=Jis an ideal ofR, andRis periodic.

As is well-known, we have

R^∼= a subdirect sum of subdirectly irreducible ringsRi. (2.4) Letσ:R→Ribe the natural homomorphism ofRontoRi. SinceRis periodic,Ri is periodic and byLemma 2.4,

Ni= the set of nilpotents ofRi=σ(N) is an ideal ofRi. (2.5) We now distinguish two cases.

Case 1 1∈/ Ri. Letxi∈Ri, and letσ:x→xi. Then byLemma 2.3,x^kis a central idempotent ofR, and hencex^k_i is a central idempotent in the subdirectly irreducible ringRi, for some positive integerk. Hencex_i^k=0 (1∈/ Ri). ThusRi=Niis a nil ring.

Case 2 1∈Ri. The above argument inCase 1shows thatx^k_i is a central idempotent in the subdirectly irreducible ringRi. Hencex^k_i =0 orx^k_i =1 for allxi∈Ri. So,Riis a local ring and for everyxi+Ni∈Ri/Ni,

xi+Ni=Ni or xi+Ni_k

=1 +Ni. (2.6)

SoRi/Niis a periodic division ring, and hence byLemma 2.2,Ri/Niis a periodic field.

(ii) Suppose R is Artinian. Using (2.3), aRis a nil right ideal for every a∈N. So, N⊆J. ButJ⊆N sinceRis Artinian. ThereforeN=J is an ideal ofR andR/N=R/J is semisimple Artinian. This implies thatR/N is isomorphic to a finite direct product R1×R2× ··· ×Rn, where eachRiis a completeti×ti matrix ring over a division ring Di. SinceRis Artinian, the idempotents ofR/Jlift to idempotents inR[2], and hence the idempotents ofR/Jare central. Iftj>1, thenE11∈Rj, and (0,..., 0,E11, 0,..., 0) is an idempotent element ofR/Jwhich is not central inR/J. This is a contradiction. Soti=1 for everyi. Therefore eachRiis a division ring andR/N is isomorphic to a finite direct

product of division rings.

The next result deals with a special kind ofD^∗-rings.

Theorem 2.10. LetRbe a ring such that every zero divisorxcan be written uniquely as x=a+e, wherea∈Nandeis idempotent.

(i) IfRis weakly periodic, thenNis an ideal ofR, andR/Nis isomorphic to a subdirect sum of fields.

(ii) IfR is Artinian, thenN is an ideal andR/N is a finite direct product of division rings.

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Proof. Lete²=e∈R,x∈R, and let f =e+ex−exe. Then f²=f and hence (e f−e)f = 0. So if f is not a zero divisor, thene f −e=0. Soe f =e, and thus f =e, which implies thatex=exe. The net result isex−exe=0 iff is not a zero divisor. Next, suppose f is a zero divisor. Then since

f =(ex−exe) +e; ex−exe∈N,eidempotent;

f =0 +f, (2.7)

it follows from uniqueness thatex−exe=0, and henceex=exein all cases. Similarly xe=exe, and thus

all idempotents ofRare central, and henceRis a normalD^∗-ring. (2.8) (i) Using (2.8),Rsatisfies all the hypotheses ofTheorem 2.9(i), and henceNis an ideal, andRis periodic. UsingLemma 2.2, for eachx∈R, there exists an integerk >1, such that x−x^k∈N, and hence

(x+N)^k=(x+N), k=k(x)>1. (2.9) By a well-known theorem of Jacobson [6], (2.9) implies thatR/Nis a subdirect sum of fields.

(ii) IfR is Artinian, then using (2.8),R satisfies the hypotheses ofTheorem 2.9(ii).

ThereforeNis an ideal andR/Nis a finite direct product of division rings.

Theorem 2.11. LetRbe a semiprimeD^∗-ring withN commutative. ThenR is either a domain or aJ-ring.

Proof. As in the proof of [3, Theorem 1] we can show that ifa^k=0, then (ar)^k=0 for all r∈R. Therefore, by Levitzki’s theorem,N= {0}. AssumeRis not a domain, and letabe any nonzero divisor of zero. Thenais potent andaRconsists of zero divisors, hence is a J-ring containinga. Therefore [ax,a]=0 for allx∈R, hence (ax)ⁿ=aⁿxⁿfor allx∈R, and alln≥2. Forxnot a zero divisor, choosen >1 such thataⁿ=aand (ax)ⁿ=ax. Then aⁿxⁿ=ax, soa(xⁿ−x)=0 andxⁿ−xis a zero divisor, hence is periodic. It follows by Chacron’s theorem thatRis a periodic ring; and sinceN= {0},Ris aJ-ring.

Example 2.12. Let

R=

0 0

,

1 1

,

1 0

0 1

,

0 1

1 0

, 0, 1∈GF(2). (2.10)

Then Ris a normal weakly periodic D^∗-ring with commuting nilpotents.R is not semiprime since the set of nilpotent elementsNis a nonzero nilpotent ideal. This example shows that we cannot drop the hypothesis “Ris semiprime” inTheorem 2.11.

InTheorem 2.14below, we study the structure of a special kind ofD^∗-rings, the class of rings in which every zero divisor is potent. Recall that a ring is semiperfect [2] if and

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only ifR/Jis semisimple (Artinian) and idempotents lift moduloJ. We need the following lemma.

Lemma 2.13. LetRbe a ring in which every zero divisor is potent. ThenN= {0}andRis normal. Moreover, IfRis not a domain, thenJ= {0}.

Proof. Ifa∈N, thenais a zero divisor and hence potent by hypothesis. Soaⁿ=afor some positive integern, and sincea∈N, there exists a positive integerksuch that 0= aⁿ^k=a. SoN= {0}. Letebe any idempotent element ofRandxis any element ofR. Then ex−exe∈N, and henceex−exe=0. Similarly,xe=exe. Soex=xeandRis normal.

Letx be a nonzero divisor of zero. ThenxJ consists of zero divisors, which are potent. ThereforexJ= {0}. But thenJconsists of zero divisors, hence potent elements, and

thereforeJ= {0}.

Theorem 2.14. Let R be a ring such that every zero divisor is potent.

(i) IfRis weakly periodic, then every element ofRis potent andRis a subdirect sum of fields.

(ii) IfRis prime, thenRis a domain.

(iii) IfRis Artinian, thenRis a finite direct product of division rings.

(iv) IfRis semiperfect, thenR/Jis a finite direct product of division rings.

Proof. (i) SinceRis weakly periodic, every elementx∈Rcan be written as

x=a+b, wherea∈N,bis potent. (2.11)

ButN= {0}(Lemma 2.13), so everyx∈Ris potent and henceRis isomorphic to a subdirect sum of fields by a well-known theorem of Jacobson.

(ii) SupposeRis a prime, thenRis a prime ring withN= {0}, and henceRis a domain.

(iii) LetR be an Artinian ring such that every zero divisor is potent. SinceN= {0} (Lemma 2.13) andRis Artinian,J=N= {0}. SoRis semisimple Artinian and hence it is isomorphic to a finite direct productR1×R2× ··· ×Rn, where eachRiis a complete ti×timatrix ring over a division ringDi. Iftj>1, thenE11∈Rj, and (0,..., 0,E11, 0,..., 0) is an idempotent element ofRwhich is not central inRcontradictingLemma 2.13. So ti=1 for everyi. Therefore eachRiis a division ring andRis isomorphic to a finite direct product of division rings.

(iv) LetR be a semiperfect ring such that every zero divisor is potent. ThenR/J is semisimple Artinian and hence it is isomorphic to a finite direct productR1×R2× ··· × Rn, where eachRi is a complete ti×ti matrix ring over a division ringDi. SinceR is semiperfect, the idempotents ofR/Jlift to idempotents inR, and hence the argument of part (iii) above implies that eachRiis a division ring andR/Jis isomorphic to a finite

direct product of division rings.

Acknowledgment

We wish to express our indebtedness and gratitude to the referee for the helpful sugges- tions and valuable comments.

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Hazar Abu-Khuzam: Department of Mathematics, American University of Beirut, Beirut 1107 2020, Lebanon

E-mail address:[email protected]

Adil Yaqub: Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA

E-mail address:[email protected]