REGULAR IFP NEAR-RINGS

A near-ring N is called a delta-near-ring if Na = N for all a is-not-an-element-of N(c), the constant part of N, and N0, the zerosymmetric part of N, is the smallest nonzero ideal of N. It is evident that every near-field is a delta-near-ring. In this paper we prove that every delta-near-ring with left identity is a near-field. The main purpose of this paper is to prove that every regular IFP near-ring is isomorphic to a subdirect product of subdirectly irreducible near-rings {N(i)} where each N(i) is either a constant near-ring or a delta-near-ring. Then many known results regarding near-rings with (P0) (i.e. for every a is-an-element-of N, there exists an integer n(a) > 1 such that a(n(a)) = a) and IFP will be corollaries to the above theorem. We also prove that every Boolean IFP near-ring is a subdirect product of near-rings {N(i)} where each N(i) is a near-ring given either in Example 3.1 or 3.3 (see below).