Abstract
A ring R is defined to be J-normal if for any \(a, r\in R\) and idempotent \(e\in R\), \(ae = 0\) implies \(Rera\subseteq J(R)\), 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 \(e\in R\) and for any \(r\in R\), \(R(1 - e)re\subseteq J(R)\) if and only if for any \(n\ge 1\), the \(n\times n\) upper triangular matrix ring \(U_{n}(R)\) is a J-normal ring if and only if the Dorroh extension of R by \({\mathbb {Z}}\) 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 \(2\times 2\) matrices over R.
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Kose, H., Kurtulmaz, Y., Ungor, B. et al. Rings having normality in terms of the Jacobson radical. Arab. J. Math. 9, 123–135 (2020). https://doi.org/10.1007/s40065-018-0231-7
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DOI: https://doi.org/10.1007/s40065-018-0231-7