Certain decompositions of matrices over abelian rings.(English)Zbl 1458.16044

Summary: A ring $$R$$ is (weakly) nil clean provided that every element in $$R$$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $$R$$ be abelian, and let $$n\in\mathbb{N}$$. We prove that $$M_n(R)$$ is nil clean if and only if $$R/J(R)$$ is Boolean and $$M_n(J(R))$$ is nil. Furthermore, we prove that $$R$$ is weakly nil clean if and only if $$R$$ is periodic; $$R/J(R)$$ is $$\mathbb{Z}_3$$, $$B$$ or $$\mathbb{Z}_3\oplus B$$ where $$B$$ is a Boolean ring, and that $$M_n(R)$$ is weakly nil clean if and only if $$M_n(R)$$ is nil clean for all $$n\geq 2$$.

MSC:

 16U99 Conditions on elements 16S50 Endomorphism rings; matrix rings 16U80 Generalizations of commutativity (associative rings and algebras)
Full Text: