Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: Exchange rings, $S$-exchange rings, clean rings, weakly clean rings, $\pi$-regular rings

MSC numbers: Primary 18G35 16G10; Secondary 18E30 16E05

Supported by: Supported by the National Science Foundation of China (Grant No. 11771212) and the National Science Foundation for Distinguished Young Scholars of Jiangsu Province (Grant No. BK2012044) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions