Rings in which every nilpotent is central

Abstract

In this paper, we introduce a class of rings in which every nilpotent element is central. This class of rings generalizes so-called reduced rings. A ring $R$ is called {\it central reduced} if every nilpotent element of $R$ is central. For a ring $R$, we prove that $R$ is central reduced if and only if $R[x_1,x_2,\ldots,x_n]$ is central reduced if and only if $R[[x_1,x_2,\ldots,x_n]]$ is central reduced if and only if $R[x_1,x_1^{-1},x_2,x_2^{-1},\ldots,x_n,x_n^{-1}]$ is central reduced. Moreover, if $R$ is a central reduced ring, then the trivial extension $T(R,R)$ is central Armendariz.