Certain decompositions of matrices over Abelian rings |
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Authors: | Nahid Ashrafi Marjan Sheibani Huanyin Chen |
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Institution: | 1.Faculty of Mathematics, Statistics and Computer Science,Semnan University,Semnan,Iran;2.Department of Mathematics,Hangzhou Normal University,Zhejiang,China |
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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 ∈ ?. 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 ?3, B or ?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. |
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