On S-exchange rings
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We introduce the concept of $S$-exchange rings to unify various subclass of exchange rings, where $S$ is a subset of the ring. Many properties on $S$-exchange rings are obtained. For instance, we show that a ring $R$ is clean if and only if $R$ is left $U(R)$-exchange, a ring $R$ is nil clean if and only if $R$ is left $(N(R)-1)$-exchange, and that a ring $R$ is $J$-clean if and only if $R$ is left $(J(R)-1)$-exchange. As a conclusion, we obtain a sufficient condition such that clean (nil clean property, respectively) can pass to corners and reprove that $J$-clean passes to corners by a different way.
Abstract
We introduce the concept of $S$-exchange rings to unify various subclass of exchange rings, where $S$ is a subset of the ring. Many properties on $S$-exchange rings are obtained. For instance, we show that a ring $R$ is clean if and only if $R$ is left $U(R)$-exchange, a ring $R$ is nil clean if and only if $R$ is left $(N(R)-1)$-exchange, and that a ring $R$ is $J$-clean if and only if $R$ is left $(J(R)-1)$-exchange. As a conclusion, we obtain a sufficient condition such that clean (nil clean property, respectively) can pass to corners and reprove that $J$-clean passes to corners by a different way.
참고문헌 (20)
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