Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 417–425

# Certain decompositions of matrices over Abelian rings

Article

## Abstract

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 ∈ ℕ. We prove that Mn(R) is nil clean if and only if R/J(R) is Boolean and Mn(J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ℤ3, B or ℤ3B where B is a Boolean ring, and that Mn(R) is weakly nil clean if and only if Mn(R) is nil clean for all n ≥ 2.

### Keywords

idempotent element nilpotent element nil clean ring weakly nil clean ring

### MSC 2010

16S34 16U10 16E50

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