J Austral Math Soc Ser A 54 pp156--168, 1993.

Minimal Ideals in Near-Rings and Localized Distributivity Conditions

Gary Birkenmeier and Henry Heatherly

Abstract

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 Î S, b₁, b₂ Î B, t Î 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 I^k is (I^m, Iⁿ)-distributive for some k, n ³ 1, m ³ 0, then either I² = 0 or I is a simple, nonnilpotent ring with every element of I distributive in R. Theorem. Let R^k be (R^m, Rⁿ)-distributive, for some k, n ³ 1, m ³ 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.

1991 AMS Subject Classification: 16A76

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Authors

Gary Birkenmeier

Henry Heatherly

University of Southwestern Louisiana, Lafayette, Louisiana 70504, U.S.A.