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Keywords:
orthogonal idempotent matrix; nilpotent matrix; matrix ring; feebly nil-clean ring
orthogonal idempotent matrix; nilpotent matrix; matrix ring; feebly nil-clean ring
Summary:
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.
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