Rings having normality in terms of the Jacobson radical

Abstract

A ring R is defined to be J-normal if for any a,r∈Ra,r∈R and idempotent e∈Re∈R, ae=0ae=0 implies Rera⊆J(R)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∈Re∈R and for any r∈Rr∈R, R(1−e)re⊆J(R)R(1−e)re⊆J(R) if and only if for any n≥1n≥1, the n×nn×n upper triangular matrix ring Un(R)Un(R) is a J-normal ring if and only if the Dorroh extension of R by ZZ 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 2×22×2 matrices over R.