STRUCTURE OF RINGS WITH CERTAIN CONDITIONS ON ZERO DIVISORS

Throughout this paper, R is an associative ring; andN ,C,C(R), and J denote, respectively, the set of nilpotent elements, the center, the commutator ideal, and the Jacobson radical. An element x of R is called potent if xn = x for some positive integer n= n(x) > 1. A ring R is called periodic if for every x in R, xm = xn for some distinct positive integers m=m(x), n = n(x). A ring R is called weakly periodic if every element of R is expressible as a sum of a nilpotent element and a potent element of R : R=N +P, where P is the set of potent elements of R. A ring R such that every zero divisor is nilpotent is called a D-ring. The structure of certain classes of D-rings was studied in [1]. Following [7], R is called normal if all of its idempotents are in C. A ring R is called a D∗-ring, if every zero divisor x in R can be written as x = a+ b, where a ∈ N , b ∈ P, and ab = ba. Clearly every D-ring is a D∗-ring. In particular every nil ring is a D∗-ring, and every domain is a D∗-ring. A Boolean ring is a D∗-ring but not a D-ring. Our objective is to study the structure of certain classes of D∗-ring.


Introduction
Throughout this paper, R is an associative ring; and N, C, C(R), and J denote, respectively, the set of nilpotent elements, the center, the commutator ideal, and the Jacobson radical.An element x of R is called potent if x n = x for some positive integer n = n(x) > 1.A ring R is called periodic if for every x in R, x m = x n for some distinct positive integers m = m(x), n = n(x).A ring R is called weakly periodic if every element of R is expressible as a sum of a nilpotent element and a potent element of R : R = N + P, where P is the set of potent elements of R. A ring R such that every zero divisor is nilpotent is called a D-ring.The structure of certain classes of D-rings was studied in [1].Following [7], R is called normal if all of its idempotents are in C. A ring R is called a D * -ring, if every zero divisor x in R can be written as x = a + b, where a ∈ N, b ∈ P, and ab = ba.Clearly every D-ring is a D * -ring.In particular every nil ring is a D * -ring, and every domain is a D * -ring.A Boolean ring is a D * -ring but not a D-ring.Our objective is to study the structure of certain classes of D * -ring.

Main results
We start by stating the following known lemmas: Lemmas 2.1 and 2.2 were proved in [5], Lemmas 2.3 and 2.4 were proved in [4].
Lemma 2.1.Let R be a weakly periodic ring.Then the Jacobson radical Let R be a periodic ring and x any element of R. Then (a) some power of x is idempotent; (b) there exists an integer n > 1 such that x − x n ∈ N.
Lemma 2.4.Let R be a periodic ring and let σ : R → S be a homomorphism of R onto a ring S. Then the nilpotents of S coincide with σ(N), where N is the set of nilpotents of R. Proof.Assume R is a D * -ring and let x be any zero divisor.Then . This implies, since x commutes with a, that (x − a) = (x − a) n = x n + sum of pairwise commuting nilpotent elements.Hence x − x n ∈ N for every zero divisor x. ( Since each such x is included in a subring of zero divisors, which is periodic by Chacron's theorem, x is periodic.
Suppose, conversely, that each zero divisor is periodic.Then by the proof of [4, Lemma 1], R is a D * -ring.
Theorem 2.7.If R is any normal D * -ring, then either R is periodic or R is a D-ring.Moreover, aR ⊆ N for each a ∈ N.
Proof.If R is a normal D * -ring which is not a D-ring, then R has a central idempotent zero divisor e.Then R = eR ⊕ A(e), where eR and A(e) both consist of zero divisors of R, hence (in view of Theorem 2.6) are periodic.Therefore R is periodic.Now consider a ∈ N and x ∈ R. Since ax is a zero divisor, hence a periodic element, (ax) j = e is a central idempotent for some j.Thus (ax) j+1 = (ax) j ax = a 2 y for some y ∈ R. Repeating this argument, one can show that for each positive integer k, there exists m such that (ax) m = a 2 k w for some w ∈ R. Therefore aR ⊆ N.
Therefore commutativity follows from Theorem 2.7 and a theorem of Herstein.Now, we prove the following result for D * -rings.This implies, using Lemma 2.1, that N = J is an ideal of R, and R is periodic.
As is well-known, we have R ∼ = a subdirect sum of subdirectly irreducible rings R i . (2.4) Let σ : R → R i be the natural homomorphism of R onto R i .Since R is periodic, R i is periodic and by Lemma 2.4, (2.5) We now distinguish two cases.
Case 1 1 / ∈ R i .Let x i ∈ R i , and let σ : x → x i .Then by Lemma 2.3, x k is a central idempotent of R, and hence x k i is a central idempotent in the subdirectly irreducible ring R i , for some positive integer k.Hence Case 2 1 ∈ R i .The above argument in Case 1 shows that x k i is a central idempotent in the subdirectly irreducible ring R i .Hence x k i = 0 or x k i = 1 for all x i ∈ R i .So, R i is a local ring and for every x i + N i ∈ R i /N i , (2.6)So R i /N i is a periodic division ring, and hence by Lemma 2.2, R i /N i is a periodic field.
(ii) Suppose R is Artinian.Using (2.3), aR is a nil right ideal for every a ∈ N. So, where each R i is a complete t i × t i matrix ring over a division ring D i .Since R is Artinian, the idempotents of R/J lift to idempotents in R [2], and hence the idempotents of R/J are central.If t j > 1, then E 11 ∈ R j , and (0,...,0,E 11 ,0,...,0) is an idempotent element of R/J which is not central in R/J.This is a contradiction.So t i = 1 for every i.Therefore each R i is a division ring and R/N is isomorphic to a finite direct product of division rings.
The next result deals with a special kind of D * -rings.
Theorem 2.10.Let R be a ring such that every zero divisor x can be written uniquely as x = a + e, where a ∈ N and e is idempotent.
(i) If R is weakly periodic, then N is an ideal of R, and R/N is isomorphic to a subdirect sum of fields.(ii) If R is Artinian, then N is an ideal and R/N is a finite direct product of division rings.
Proof.Let e 2 = e ∈ R, x ∈ R, and let f = e + ex − exe.Then f 2 = f and hence (e f − e) f = 0.So if f is not a zero divisor, then e f − e = 0.So e f = e, and thus f = e, which implies that ex = exe.The net result is ex − exe = 0 if f is not a zero divisor.Next, suppose f is a zero divisor.Then since f = (ex − exe) + e; ex − exe ∈ N, e idempotent; it follows from uniqueness that ex − exe = 0, and hence ex = exe in all cases.Similarly xe = exe, and thus all idempotents of R are central, and hence R is a normal D * -ring.
(2.8) (i) Using (2.8), R satisfies all the hypotheses of Theorem 2.9(i), and hence N is an ideal, and R is periodic.Using Lemma 2.2, for each x ∈ R, there exists an integer k > 1, such that x − x k ∈ N, and hence (2.9) By a well-known theorem of Jacobson [6], (2.9) implies that R/N is a subdirect sum of fields.
(ii) If R is Artinian, then using (2.8), R satisfies the hypotheses of Theorem 2.9(ii).Therefore N is an ideal and R/N is a finite direct product of division rings.
Theorem 2.11.Let R be a semiprime D * -ring with N commutative.Then R is either a domain or a J-ring.
Proof.As in the proof of [3,Theorem 1] we can show that if a k = 0, then (ar) k = 0 for all r ∈ R. Therefore, by Levitzki's theorem, N = {0}.Assume R is not a domain, and let a be any nonzero divisor of zero.Then a is potent and aR consists of zero divisors, hence is a J-ring containing a. Therefore [ax,a] = 0 for all x ∈ R, hence (ax) n = a n x n for all x ∈ R, and all n ≥ 2. For x not a zero divisor, choose n > 1 such that a n = a and (ax) n = ax.Then a n x n = ax, so a(x n − x) = 0 and x n − x is a zero divisor, hence is periodic.It follows by Chacron's theorem that R is a periodic ring; and since Then R is a normal weakly periodic D * -ring with commuting nilpotents.R is not semiprime since the set of nilpotent elements N is a nonzero nilpotent ideal.This example shows that we cannot drop the hypothesis "R is semiprime" in Theorem 2.11.
In Theorem 2.14 below, we study the structure of a special kind of D * -rings, the class of rings in which every zero divisor is potent.Recall that a ring is semiperfect [2] if and only if R/J is semisimple (Artinian) and idempotents lift modulo J.We need the following lemma.
Lemma 2.13.Let R be a ring in which every zero divisor is potent.Then N = {0} and R is normal.Moreover, If R is not a domain, then J = {0}.
Proof.If a ∈ N, then a is a zero divisor and hence potent by hypothesis.So a n = a for some positive integer n, and since a ∈ N, there exists a positive integer k such that 0 = a n k = a.So N = {0}.Let e be any idempotent element of R and x is any element of R. Then ex − exe ∈ N, and hence ex − exe = 0. Similarly, xe = exe.So ex = xe and R is normal.
Let x be a nonzero divisor of zero.Then xJ consists of zero divisors, which are potent.Therefore xJ = {0}.But then J consists of zero divisors, hence potent elements, and therefore J = {0}.Theorem 2.14.Let R be a ring such that every zero divisor is potent.
(i) If R is weakly periodic, then every element of R is potent and R is a subdirect sum of fields.
Proof.(i) Since R is weakly periodic, every element x ∈ R can be written as x = a + b, where a ∈ N, b is potent. (2.11) But N = {0} (Lemma 2.13), so every x ∈ R is potent and hence R is isomorphic to a subdirect sum of fields by a well-known theorem of Jacobson.
(ii) Suppose R is a prime, then R is a prime ring with N = {0}, and hence R is a domain.
(iii) Let R be an Artinian ring such that every zero divisor is potent.Since N = {0} (Lemma 2.13) and R is Artinian, J = N = {0}.So R is semisimple Artinian and hence it is isomorphic to a finite direct product R 1 × R 2 × ••• × R n , where each R i is a complete t i × t i matrix ring over a division ring D i .If t j > 1, then E 11 ∈ R j , and (0,...,0,E 11 ,0,...,0) is an idempotent element of R which is not central in R contradicting Lemma 2.13.So t i = 1 for every i.Therefore each R i is a division ring and R is isomorphic to a finite direct product of division rings.
(iv) Let R be a semiperfect ring such that every zero divisor is potent.Then R/J is semisimple Artinian and hence it is isomorphic to a finite direct product R 1 × R 2 × ••• × R n , where each R i is a complete t i × t i matrix ring over a division ring D i .Since R is semiperfect, the idempotents of R/J lift to idempotents in R, and hence the argument of part (iii) above implies that each R i is a division ring and R/J is isomorphic to a finite direct product of division rings.
Definition 2.5.A ring is said to be a D-ring if every zero divisor is nilpotent.A ring R is called a D * -ring if every zero divisor x in R can be written as x = a + b, where a ∈ N, b ∈ P, and ab = ba.