Certain decompositions of matrices over Abelian rings

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 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 ℤ 3 , B or ℤ 3 ⊕ 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 ≥ 2.