On idempotents in relation with regularity
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We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring $R$ is said to be {\it right attaching-idempotent} if for $a\in R$ there exists $0\neq b\in R$ such that $ab$ is an idempotent.Next $R$ is said to be {\it generalized regular} if for $0\neq a\in R$ there exist nonzero $b\in R$ such that $ab$ is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.
Abstract
We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring $R$ is said to be {\it right attaching-idempotent} if for $a\in R$ there exists $0\neq b\in R$ such that $ab$ is an idempotent.Next $R$ is said to be {\it generalized regular} if for $0\neq a\in R$ there exist nonzero $b\in R$ such that $ab$ is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.
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