ON THE MAXIMAL IDEALS OF NON-ZERO-SYMMETRIC NEAR-RINGS AND OF COMPOSITION ALGEBRAS OF POLYNOMIAL FUNCTIONS ON Ω-GROUPS

Abstract

Let G be a finite group, and let (P(G); +, o) be the near-ring of all unary polynomial functions on G. We describe the maximal ideals of this near-ring. Our approach also allows us to determine the maximal congruences of the composition algebra of polynomial functions on a finite Σ-group. We apply these results to find the maximal ideals of a finite near-ring with 1 that is faithful on its constants. In some occasions, the intersection of the maximal ideals yields an ideal that has properties that are similar to nilpotency.