Certain decompositions of matrices over Abelian rings
- First Online:
- 31 Downloads
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 ℤ3 ⊕ B 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.
Keywordsidempotent element nilpotent element nil clean ring weakly nil clean ring
MSC 201016S34 16U10 16E50
Unable to display preview. Download preview PDF.
- D. Andrica, G. Călugăreanu: A nil-clean 2×2 matrix over the integers which is not clean. J. Algebra Appl. 13 (2014), Article ID 1450009, 9 pages.Google Scholar
- S. Breaz, P. Danchev, Y. Zhou: Rings in which every element is either a sum or a difference of a nilpotent and an idempotent. J. Algebra Appl. 15 (2016), Article ID 1650148, 11 pages.Google Scholar
- H. Chen: Rings Related to Stable Range Conditions. Series in Algebra 11, World Scientific, Hackensack, 2011.Google Scholar
- W. W. McGovern, S. Raja, A. Sharp: Commutative nil clean group rings. J. Algebra Appl. 14 (2015), Article ID 1550094, 5 pages.Google Scholar