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 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 ℤ3, B or ℤ3 ⊕ 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.
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Huanyin Chen was supported by the Natural Science Foundation of Zhejiang Province, China (No. LY17A010018).
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Ashrafi, N., Sheibani, M. & Chen, H. Certain decompositions of matrices over Abelian rings. Czech Math J 67, 417–425 (2017). https://doi.org/10.21136/CMJ.2017.0677-15
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DOI: https://doi.org/10.21136/CMJ.2017.0677-15