Geodesic
On feebly nil-clean rings
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 87-94
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A ring $R$ is feebly nil-clean if for any $a\in R$ there exist two orthogonal idempotents $e,f\in R$ and a nilpotent $w\in R$ such that $a=e-f+w$. Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.
A ring $R$ is feebly nil-clean if for any $a\in R$ there exist two orthogonal idempotents $e,f\in R$ and a nilpotent $w\in R$ such that $a=e-f+w$. Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.
DOI : 10.21136/CMJ.2023.0215-22
Classification : 15A23, 15B33, 16U99
Keywords: orthogonal idempotent matrix; nilpotent matrix; matrix ring; feebly nil-clean ring 
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Sheibani Abdolyousefi, Marjan; Pouyan, Neda. On feebly nil-clean rings. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 87-94. doi: 10.21136/CMJ.2023.0215-22

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