Rings having normality in terms of the Jacobson radical

Abstract

A ring R is defined to be J-normal if for any a, r∈ R and idempotent e∈ R, ae= 0 implies Rera⊆ J(R) , where J(R) is the Jacobson radical of R. The class of J-normal rings lies between the classes of weakly normal rings and left min-abel rings. It is proved that R is J-normal if and only if for any idempotent e∈ R and for any r∈ R, R(1 - e) re⊆ J(R) if and only if for any n≥ 1 , the n× n upper triangular matrix ring Un(R) is a J-normal ring if and only if the Dorroh extension of R by Z is J-normal. We show that R is strongly regular if and only if R is J-normal and von Neumann regular. For a J-normal ring R, it is obtained that R is clean if and only if R is exchange. We also investigate J-normality of certain subrings of the ring of 2 × 2 matrices over R.