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.

中文翻译：

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就Jacobson根而言具有正态性的环

甲环- [R被定义为J-正常的，如果对于任何\（A，R \中的R \）和幂等\（E \中的R \） ，\（AE = 0 \）意味着\（RERA \ subseteqĴ（R ）\），其中Ĵ（[R ）是Jacobson根的ř。J正态环的类别介于弱正态环和左min-abel环之间。实践证明，- [R是J-正常当且仅当对于任何幂等\（E \中的R \）和任何\（R \中的R \） ，\（R（1 - E）重新\ subseteqĴ（R ）\）当且仅当对于任何\（N \ GE 1 \） ，该\（N \ n次\）当且仅当R由\（{\ mathbb {Z}} \）的Dorroh扩展是J-normal时，上三角矩阵环\（U_ {n}（R）\）是J-normal环。我们证明，当且仅当R是J正态且冯·诺依曼正则时，R才是强规则的。对于J正态环R，当且仅当R是交换时，获得R是清洁的。我们还研究R上\（2×2）矩阵环的某些子环的J正规性。