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Minimal ideals in near-rings and localized distributivity conditions

Published online by Cambridge University Press:  09 April 2009

Gary Birkenmeier  and
Henry Heatherly
Gary Birkenmeier
Affiliation:
University of Southwestern Louisiana Lafayette, Louisiana 70504, U.S.A.
Henry Heatherly
Affiliation:
University of Southwestern Louisiana Lafayette, Louisiana 70504, U.S.A.
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Abstract

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Let B, S, and T be subsets of a (left) near-ring R with B and T nonempty. We say B is (S, T)-distributive if s(b1+b2)t = sb1t + sb2t, for each s ∈ S, b1, 2 ∈ B, t ∈ T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals. Theorem. If I is a minimal ideal of R and Ik is (Im, In)-distributive for some k, n ≧ 1, m ≧ 0, then either I2 = 0 or I is a simple, nonnilpotent ring with every element of I distributive in R. Theorem. Let Rk be (Rm, Rn)-distributive, for some k, n ≧ 1, m ≧ 0; if R is semiprime or is a subdirect product of simple near-rings, then R is a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. If R → A → 0 is an exact sequence of near-rings, then conditions on A are given which will impose conditions on the minimal ideals of R.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 2 , April 1993 , pp. 156 - 168
Copyright
Copyright © Australian Mathematical Society 1993

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