Certain Near-Rings are Rings

The aim of this paper is to generalize the results in [1] and [13]. Here, we are interested in two problems concerning certain classes of near rings satisfying one of the following polynomials identities : (*) For each x, y in a near-ring N , there exist positive integers t = (x, y) ≥ 1 and s = s (x, y) > 1 such that xy = ±y s x t. (* *) For each x, y in a near-ring N , there exist positive integers t = t (x, y) ≥ 1 and s = s (x, y) > 1 such that xy = ±x t y s .

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“…Our definitions are as in [2]. A near-ring ideal P will be called completely prime if ab € P implies a € P or b € P .…”

Section: Preliminariesmentioning

confidence: 99%

“…Part of the recent work on near-rings has been concerned with sufficient conditions for near-rings to be commutative (see [7], [2], [6], [JO]). Recently Bell [2J showed that if if is a distributively generated (d.g.)…”

Section: Introductionmentioning

confidence: 99%

On the commutativity of near-rings III

Ligh

1972

Bull. Austral. Math. Soc.

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Part of the recent work on near-rings has been concerned with sufficient conditions for near-rings to be commutative.Recently Howard E. Bell proved that if a d.g. near-ring i?has an identity and for each x, y in R , there exists an w(x, y) > 1 > such that (xy-yx) = xy -yx , then if is a commutative ring. In this paper we drop the requirement that R has an identity and show that the other condition is sufficient (and necessary) for R to be commutative. The inspiration for an important lemma comes from a result of B.H. Neumann.

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Certain Near-Rings are Rings

Cited by 22 publications

References 3 publications

Certain near‐rings are rings, II

Certain near‐rings are rings, II

On Structure and Commutativity of Near - Rings

On the commutativity of near-rings III

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