An Elaboration of Annihilators of Polynomials

Abstract

Abstract. In this note we elaborate first on well-known theorems for annihilators of polynomials over IFP rings by investigating the concrete shapes of nonzero constant annihilators. We consider next a generalization of IFP which preserves Abelian property, in relation with annihilators of polynomials, observing the basic structure of rings satisfying such condition. Annihilators of polynomials on IFP ringsIFP and Abelian ring property have important roles in noncommutative ring theory and module theory. We continue in this section the studies of Nielsen [22] and Shin [25], being concerned with the constant annihilators of polynomials, and introduce a generalization of IFP which preserves Abelian property.Throughout this note every ring is an associative ring with identity unless otherwise stated. Given a ring R, let N (R), N * (R), and J(R) denote the set of all nilpotent elements, the prime radical, and the Jacobson radical in R, respectively. The polynomial (resp., power series) ring with an indeterminate x over R is denoted by R[x] (resp., R[[x]]). The right annihilator of S in R is denoted by r R (S), and by r R (a) when S = {a}. The degree of a polynomial f (x) is denoted by deg f (x). The n by n full (resp. upper triangular) matrix ring over R is denoted by Mat n (R) (resp. U n (R)), and denote by e ij the matrix with (i, j)-entry 1 and elsewhere zero. Z denotes the ring of integers, and Z n denotes the ring of integers modulo n.A ring R (possibly without identity) is called reduced if N (R) = 0. A wellknown property that unifies the commutativity and the reduced condition is the insertion-of-factors-property. Due to Bell [4], a ring R (possibly without identity) is called to satisfy the insertion-of-factors-property (simply, an IFP ring) if ab = 0 implies aRb = 0 for a, b ∈ R. Narbonne [21] and Shin [25] used the terms semicommutative and SI for the IFP, respectively. Commutative rings are clearly IFP, and any reduced ring is IFP by a simple computation.