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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)
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