A WEAK PERIODICITY CONDITION FOR RINGS

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.


Introduction
An element x of the ring R is called periodic if there exist distinct positive integers m, n such that x m = x n ; and x is potent if there exists n > 1 for which x n = x.We denote the set of potent elements by P or P(R), the set of nilpotent elements by N or N(R), the center by Z or Z(R), and the Jacobson radical by J or J(R).
The ring R is called periodic if each of its elements is periodic, and R is called weakly periodic if R = 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 periodic rings have nice commutativity behavior; in particular, Herstein [10] showed that if R is periodic and N ⊆ Z, then R is commutative-a result which extends easily to weakly periodic rings.Various generalized periodic and weakly periodic rings have been introduced in recent years, and their commutativity behavior has been explored [6,7,13,14].
Define R to be semi-weakly periodic if R\(J ∪ Z) ⊆ P + N. Clearly the class of semiweakly periodic rings is quite large; it contains all 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.

Preliminaries
We fix some more notation.If x, y ∈ R, the symbol [x, y] denotes the commutator xy − yx; and if S,T ⊆ R, [S,T] denotes the set {[s, t]/s ∈ S, t ∈ T}.For x ∈ R, the symbol x denotes the subring generated by x; and the symbols C(R) and E stand for the commutator ideal and the set of idempotents of R. If ℘ is a ring property, a ring R having the property is called a ℘-ring.
We also state some known results we require, the first of which is trivial.
Lemma 2.1.If R is any ring and S is any proper additive subgroup of R, the centralizer of R\S is equal to Z(R).
Lemma 2.2 [9].If R is a ring such that for each x ∈ R, there exists an integer n > 1 such that Lemma 2.4 [8].Let R be a ring such that for each x ∈ R, there exist a positive integer m and a polynomial p(X) with integer coefficients for which x m = x m+1 p(x).Then R is periodic.Theorem 2].Let R be an arbitrary ring, and let We conclude this section with a theorem stating some basic results on semi-weakly periodic rings.Proof.(a) Let I be an ideal of R and x ∈ I\(J(I) ∪ Z(I)).Clearly x / ∈ Z(R); and since Therefore, x = a + u, where u ∈ N and a ∈ P; and we may choose n > 1 such that u n = 0 and a n = a.It follows that a = a n = (x − u) n ∈ I and hence u ∈ I. Consequently, I is semi-weakly periodic.
(b) Let S be a ring and let ϕ : R → S be an epimorphism.Let y ∈ S\(J(S) ∪ Z(S)), and let (d) Clearly, if Z is periodic and J is nil, then R is weakly periodic.Conversely, suppose R is weakly periodic.By the argument in the proof of (a), J is weakly periodic; and since J contains no nonzero idempotents and hence no nonzero potent elements, J is nil.Let z ∈ Z and write z = a + u with u ∈ N and a n = a, n > 1.Then [a,u] = 0 and hence z − z n is a sum of commuting nilpotent elements, so that z − z n ∈ N. It follows from Lemma 2.4 that Z is periodic.
(e) Since N ⊆ Z, N is an ideal; and by (c), for each x ∈ R\(J ∪ Z), there exists n > 1 for which x − x n ∈ N ⊆ Z. Since J ⊆ Z, Lemma 2.2 now implies that R is commutative.

Commutativity results
It is proved in [2] that if R is periodic and N is commutative, then N is an ideal.Surprisingly, this result extends to semi-weakly periodic rings.Theorem 3.1.Let R be a semi-weakly periodic ring with R = J.If N is commutative, then N is an ideal.
Proof.Since N is commutative, N is an additive subgroup and is closed under multiplication.By Lemma 2.5, PN ⊆ N; and it follows that (R\(J ∪ Z))N ⊆ N. Since ZN ⊆ N, we have (R\J)N ⊆ N. Now let u ∈ N and y ∈ J, and let x ∈ R\J.Then x + y ∈ R\J, and hence Proof.If R = J, then R is commutative and the conclusion is immediate.If R = J, N is an ideal by Theorem 3.1 and hence N ⊆ J. Therefore, in R/J every element is either potent or central, so that R/J is commutative by Lemma 2.2.Our result now follows from Lemma 2.3.
Herstein's theorem on commutativity of periodic rings with N ⊆ Z also has an extension to semi-weakly periodic rings.
Proof.Since N ⊆ Z, N is an ideal and hence N ⊆ J.By Theorem 2.6(c), for each x ∈ R\(J ∪ Z), there exists n > 1 such that x n − x ∈ N ⊆ Z; moreover, since ex − exe and xe − exe are in N for all x ∈ R and all e ∈ E, we can use a standard argument to show that E ⊆ Z.
By Theorem 2.6(e), we need only show that J ⊆ Z. Assume first that R has 1, and suppose that w ∈ 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 that q(1 − w) ∈ J and at most one prime q such that q(1 − w) ∈ Z, hence there exists a prime p such that p(1 − w) / ∈ J ∪ Z.Thus, there exists and this fact, together with the fact that (1 − w) n − (1 − w) ∈ N, shows that w is finite and hence w is periodic.But the only periodic elements in J are nilpotent, so we have contradicted our hypothesis that w ∈ J\Z.Thus, J ⊆ Z as required.Now suppose R does not have 1.If R = J ∪ Z, then we must have R = Z, since a group cannot be the union of two proper subgroups; therefore we may suppose that R = J ∪ Z, in which case P = {0}, since N ⊆ J. Let a ∈ P\{0} with a n = a, n > 1.Then e = a n−1 is an idempotent, necessarily central; and by Theorem 2.6(a), eR is semi-weakly periodic with multiplicative identity e and nilpotent elements central.Hence, by the argument above, Our next theorem may be regarded as an extension of [3,Theorem 2].Theorem 3.4.Let R be a semi-weakly periodic ring with R = J.If N is commutative and each element of R\(J ∪ Z) is uniquely expressible as a sum of a potent element and a nilpotent element, then R is commutative.
Proof.We begin by showing that E ⊆ Z-a fact which will enable us to pass from the case of R with 1 to the general case by an argument similar to that used in the proof of Theorem 3.3.
Suppose e ∈ E\Z.Then if [e,x] = 0, either ex − exe = 0 or xe − exe = 0; and we assume ex − exe = 0, in which case the nonzero idempotent f = e + ex − exe is not in J ∪ Z. Then we have (e + ex − exe) + 0 = e + (ex − exe)-two representations of f 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 that R has 1.In view of Theorem 3.3, we need only show that N ⊆ Z. Suppose that w ∈ N\Z.The same argument used in the previous proof shows that (R,+) is a torsion group and there exists n > 1 such that (1 + w) n − (1 + w) ∈ N; and it follows that 1 + w is finite.Thus, 1 + w is 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 in P, and 0 and w are in N; therefore w = 0, contradicting our assumption that w / ∈ 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 for R with 1.This observation suggests the following general theorem.Theorem 3.5.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.If R is any semi-weakly periodic ℘-ring in which E ⊆ Z and J is commutative, then R is commutative.
Proof.If R = J ∪ Z, then R = J or R = Z and hence R is commutative.Otherwise, if a ∈ P\{0}, let a n = a and a n−1 = e.Then e is a central idempotent; and eR is commutative, since it is a semi-weakly periodic ℘-ring with 1. Thus [ea,ex] = 0 for all x ∈ R, that is, a n−1 [a,x] = 0 = [a,x] for all x ∈ R. Therefore, P ⊆ Z and in particular [P,J] = 0. Since N ⊆ J, [N,J] = 0 and we conclude that [R\J,J] = 0, so that J ⊆ Z by Lemma 2.1.Commutativity of R now follows from Theorem 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.then R is commutative.
Proof.Let ℘ be the following property: R is n(n − 1)-torsion-free with J commutative and R satisfies ( * ).It is proved in [5] that any n(n − 1)-torsion-free ring with 1 which satisfies ( * ) is commutative.Hence, our theorem follows from Theorem 3.5 once we show that property ℘ forces N to be an ideal.In fact, we show that ( * ) together with the hypothesis that J is commutative forces N to be an ideal.Consider R/J, which is a subdirect product of primitive rings R α satisfying ( * ).Since ( * ) is inherited by subrings and epimorphic images, it follows from Jacobson's density theorem that either all R α are division rings, or there exist a division ring D and an integer m > 1 for which the ring of m × m matrices over D satisfies ( * ).A simple substitution shows that the second alternative cannot occur, so each R α is a division ring such that (xy) n = x n y n for all x, y ∈ R α .But such division rings are commutative by a well-known theorem of Herstein [11,12], so R/J is commutative and hence N is an ideal by Lemma 2.3.

Lemma 2 . 3 .
If R has an ideal I such that I and R/I are both commutative, then N is an ideal and C(R) ⊆ N. Proof.Since R/I is commutative, [x, y] ∈ I for all x, y ∈ R; and since I is commutative, R satisfies the polynomial identity [[x, y],[z,w]] = 0, which is not satisfied by the ring of 2 × 2 matrices over any GF(p).The result now follows by [1, Theorem 1].

Theorem 2 . 6 .
Let R be a semi-weakly periodic ring.(a) Every ideal of R is semi-weakly periodic.(b) Every epimorphic image of R is semi-weakly periodic.(c) If N is an ideal, then for each x ∈ R\(J ∪ Z), there exists n > 1 for which x − x n ∈ N. (d) R is weakly periodic if and only if Z is periodic and J is nil.(e) If N ⊆ J ⊆ Z, then R is commutative.

Theorem 3 . 6 .
Let n > 1 be a fixed positive integer and let R be an n(n − 1)-torsion-free semi-weakly periodic ring with E ⊆ Z and J commutative.If ( * ) (xy) n = x n y n ∀x, y ∈ R\N, (3.1)