Permutation identity near-rings and ?localized? distributivity conditions

Abstract

A near-ringR is said to satisfy apermutation identity if there is some non-identity permutation σ of lengthn such that Πaj=Πaσ(j), for eacha1, ...,an∈R. Numerous examples of permutation identity near-rings are given. The theory is then developed making use of various “localized” distributive conditions, which include as special cases most of the standard global ones (e. g., d. g., pseudo-distributive). These localized conditions only assume distributivity among the elements of certain special (and often “small”) sets. Particularly useful for such sets are powers of the ideals generated by the sets of Lie commutators, additive commutators, or distributive elements. Examples are given where a localized condition holds yet none of the usual global ones do. Results are obtained concerning prime, semiprime, or maximal ideals as well as regular, simple, or subdirectly irreducible near-rings.