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

TL;DR

This paper introduces CN-rings, where every element decomposes into a central and a nilpotent part, characterizes such decompositions in matrix rings, and explores conditions affecting their structure.

Contribution

It defines CN-rings, characterizes elements with CN-decompositions in matrix rings, and analyzes structural conditions for subrings over CN rings.

Findings

$M_n(F)$ is not a CN-ring for any field $F$

If $M_n(D)$ is a CN-ring over a division ring $D$, then the center's cardinality exceeds $n$

Certain subrings of matrix rings over CN rings are also CN under specific conditions

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$CN$ 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.