On $JR$-rings

Document Type : Original Article


Department of Mathematics, University of Allahabad, Prayagraj-211002, India.



We introduce the idea of a $JR$-ring. The class of $JR$-rings contains properly regular rings, local rings, $J$-clean rings and $P$-clean rings. In support, we provide some examples and counter examples. We establish some extensions of $JR$-rings, and show that $R$ is a $JR$-ring if and only if $K_0(R)$ is a $JR$-ring. A ring with the only idempotents $0$ and $1$ is a $JR$-ring if and only if it is a local ring. If $R$ has no nonzero idempotents, then $R$ is a $J$-clean ring if and only if $R$ is a $JR$-ring. If $R$ is $J$-semisimple and left or right quasi-duo ring, then $R$ is a $JR$-ring if and only if $R$ is a $NR$-clean ring.