Certain near‐rings are rings, II

Abstract

ABSTRACT. We investigate distributively-generated near-rings R which satisfy one of the following conditions:(i) for each x,y e R, there exist positive integers m, n for which xy ymxn; (ii) for each x,y e R, there exists a positive integer n such that xy (yx) n. Under appropriate additional hypotheses, we prove that R must be a commutative ring.

Certain conditions under which near-rings are rings

“…Besides providing a simpler and more attractive proof of a result due to Bell [5], our theorem generalises the results proved in [1], [2] and [10].…”

“…D Now in view of [4], Lemma 1 our lemma at once yields that N is a two-sided ideal which in turn, together with the main theorem of [4], proves the following: THEOREM . Let R be a d-g near-ring satisfying any one of the conditions (l)- (5). Then R is commutative.…”

“…U COROLLARY 2 . Let R be a d-g near-ring with unity satisfying any one of the conditions (l)- (5). Then R is a commutative ring.…”

On some properties of near-rings

“…Studying when a near-ring is a ring or a commutative ring is one of the most important questions in the theory of near-rings, which has been the subject of many investigations since the 70s of the foregoing century. For example, among many others, Ligh in [11] and Bell in [2][3][4] have established some conditions that force a near-ring to be a ring. Later on, Bell and Mason in [7] have studied many properties of 3-prime near-rings that make them commutative rings.…”