INSERTION-OF-FACTORS-PROPERTY WITH FACTORS

Abstract. We in this note study a ring theoretic property which unifies Armendariz and IFP. We call this new concept INFP. We first show that idempotents and nilpotents are connected by the Abelian ring property. Next the structure of INFP rings is studied in relation to several sorts of algebraic systems. 1. INFP rings Throughout this note every ring is an associative ring with identity unless otherwise stated. Given a ringR, let I(R) andN(R) denote the set of all idempotents and the set of all nilpotent elements in R, respectively. A nilpotent elements is also called a nilpotent simply. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). Use eij for the matrix with (i, j)-entry 1 and elsewhere 0. R[x] denotes the polynomial ring with an indeterminate x over R. For f(x) ∈ R[x], let Cf(x) denote the set of all coefficients of f(x). Following the literature, the index of nilpotency of a ∈ N(R) is the least positive integer n such that an = 0, write i(a) for n. The index of nilpotency of a subset S of R is the supremum of the indices of nilpotenc