Minimal ideals in near-rings and localized distributivity conditions

Abstract

AbstractLetB, S, andTbe subsets of a (left) near-ringRwithBandTnonempty. We sayBis (S, T)-distributiveifs(b1+b2)t=sb1t+sb2t, for eachs∈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. IfIis a minimal ideal ofRandIkis (Im,In)-distributive for somek,n≧ 1,m≧ 0, then eitherI2= 0 orIis a simple, nonnilpotent ring with every element ofIdistributive inR. Theorem. LetRkbe (Rm,Rn)-distributive, for somek,n≧ 1, m ≧ 0; ifRis semiprime or is a subdirect product of simple near-rings, thenRis a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. IfR→A→ 0 is an exact sequence of near-rings, then conditions onAare given which will impose conditions on the minimal ideals ofR.