Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 417–425

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 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.


idempotent element nilpotent element nil clean ring weakly nil clean ring 

MSC 2010

16S34 16U10 16E50 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics, Statistics and Computer ScienceSemnan UniversitySemnanIran
  2. 2.Department of MathematicsHangzhou Normal UniversityZhejiangChina

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