Central Armendariz Rings

Abstract

We introduce the notion of central Armendariz rings which are a generalization of Armendariz rings and investigate their properties. We show that the class of central Armendariz rings lies strictly between classes of Armendariz rings and abelian rings. For a ring R, we prove that R is central Armendariz if and only if the polynomial ring R[x] is central Armendariz if and only if the Laurent polynomial ring R[x; xn] is central Armendariz. Moreover, it is proven that if R is reduced, then R[x]=(xn) is central Armendariz, the converse holds if R is semiprime, where (xn) is the ideal generated by xn and n 2. Among others we also show that R is a reduced ring if and only if the matrix ring Tn-2 n (R) is central Armendariz, for a natural number n 3 and k = [n=2].