On a generalization of McCoy rings

Victor Camillo; Tai Keun Kwak; Yang Lee

doi:10.4134/JKMS.2013.50.5.959

Rege-Chhawchharia, and Nielsen introduced the concept of right McCoy ring, based on the McCoy's theorem in 1942 for the anni- hilators in polynomial rings over commutative rings. In the present note we concentrate on a natural generalization of a right McCoy ring that is called a right nilpotent coefficient McCoyring (simply, a right NC-McCoy ring). The structure and several kinds of extensions of right NC-McCoy rings are investigated, and the structure of minimal right NC-McCoy rings is also examined. Throughout this paper R denotes an associative ring with identity unless otherwise stated. Let N(R) be the set of all nilpotent elements in R. We use R(x) to denote the polynomial ring with an indeterminate x over R. Let Cf(x) denote the set of all coefficients off(x) ∈ R(x). Denote the n by n full matrix ring over R by Matn(R) and the n by n upper triangular matrix ring over R by Un(R). Use Eij for the matrix with (i,j)-entry 1 and elsewhere 0. By Zn we mean the ring of integers modulo n. McCoy (27) showed that if two polynomials annihilate each other over a commutative ring, then each polynomial has a nonzero annihilator in the base ring. Weiner (16) showed this fact fails in non-commutative rings. Based on this result, Nielsen (29) and Rege-Chhawchharia (30) each called a non-commutative ring R right McCoy (resp., left McCoy) if whenever any nonzero polynomials f(x),g(x) ∈ R(x) satisfy f(x)g(x) = 0, then f(x)c = 0 (resp., cg(x) = 0) for some nonzero c ∈ R, and a ring R is called McCoy if it is both left and right McCoy. Rege-Chhawchharia also called R an Armendariz ring (30, Definition 1.1) if whenever any polynomials f(x),g(x) ∈ R(x) satisfy f(x)g(x) = 0, then ab = 0 for each a ∈ Cf(x) and b ∈ Cg(x). Any reduced ring (i.e., it has no nonzero nilpotent elements) is Armendariz by (4, Lemma 1). Armendariz rings are clearly McCoy but the converse does not hold by (30, Remark 4.3). A ring is called Abelian if every idempotent is central. Armendariz rings are Abelian by the proof of (2, Theorem 6).

On a generalization of McCoy rings

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