Rings in which elements are a sum of a central and a nilpotent element

Yosum Kurtulmaz; Abdullah Harmancı

paper

Rings in which elements are a sum of a central and a nilpotent element

arXiv:2005.12575

Abstract

In this paper, we introduce a new class of rings whose elements are a sum of a central element and a nilpotent element, namely, a ring $R$ is called $C N$ if each element $a$ of $R$ has a decomposition $a = c + n$ where $c$ is central and $n$ is nilpotent. In this note, we characterize elements in $M_{n} (R)$ and $U_{2} (R)$ having CN-decompositions. For any field $F$ , we give examples to show that $M_{n} (F)$ can not be a CN-ring. For a division ring $D$ , we prove that if $M_{n} (D)$ is a CN-ring, then the cardinality of the center of $D$ is strictly greater than $n$ . Especially, we investigate several kinds of conditions under which some subrings of full matrix rings over CN rings are CN.

15 pages