On commutativity of periodic rings and near-rings

Abstract

A ring or near-ring R is called periodic if for each xCR, there exist distinct positive integers n, m for which x"=x". This paper continues the study of commutativity in such rings and near-rings which was previously undertaken in [1]--[5]. The first section is devoted to new proofs of two basic results Chacron's result that co-algebraic rings are necessarily periodic, and Herstein's result that periodic rings with central nilpotent elements are commutative. The next section deals with commutativity of periodic rings with constraints on certain commutators involving nilpotent elements, and the third is a discussion of structure and commutativity of certain of the 6-rings introduced by PtJWCrIA and YAQUB in [17]. The final section presents related results on commutativity of periodic near-rings. Throughout the paper Z will denote the ring of integers, Z + the set of positive integers, and Z[X] the ring of polynomials in one indeterminate with integer coefficients. The (multiplicative) center of R will be denoted by C, the set of nilpotent elements by N, and the additive group by (R, +). For each subset S of R, the left, right and two-sided annihilators will be denoted respectively by At(S),Ar(S), and A(S); the subring or sub-near-ring generated by S will be denoted by (S). As usual, the symbol [x, y] will stand for the commutator xy -yx ; and the symbol Cg(R) will indicate the commutator ideal of R. Generalized commutators are defined by letting Ix, y]l=[x, y] and extending inductively by the formula [x, y],=[[x, y],_l, y], n~_2. We shall make frequent use of the following properties of periodic rings [4, Lemma 1]: (Px) For each xCR, some power of x is idempotent. (P2) For each xER, there exists an integer n (x )> l for which x-x"(X)EN. (Ps) Each xCR can be expressed in the form y+w, where w~N and y"=y for some n=n(y)>l. (P4) If I is an ideal of R and x + I is a non-zero nilpotent element of R/I, then R contains a nilpotent element u such that u=-x (rood/).