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

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.

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