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

2020/05/26 by Yosum Kurtulmaz, Kurtulmaz, Yosum, Abdullah Harmancı +1
Mathematics · #15B33 #15B36 #16S70 #16U99 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2005.12575

openalex publication_date 2020/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 calledCN 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 U₂(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.

Citations

• Nil clean rings

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