Let J(R) denote the Jacobson radical of a ring R. R is called J-semicommutative if for any \(a,b\in R, ab=0\) implies \(aRb\subseteq J(R)\). We observe that the class of J -semicommutative rings contains the class of left (right) quasi-duo rings and various existing versions of semicommutative rings, symmetric rings and reversible rings. We provide some conditions for J-semicommutative rings to be left quasi-duo. Finally, the consequences of J-semicommutativity conditions over some classes of rings are discussed.

中文翻译：

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重温 J-半交换环

令J ( R ) 表示环R的雅各布森根。如果对于R中的任何\(a,b\)，ab=0\)意味着\(aRb\subseteq J(R)\) ，则R称为J半交换。我们观察到J-半交换环类包含左（右）拟二重环类以及半交换环、对称环和可逆环的各种现有版本。我们提供了J半交换环成为拟二重环的一些条件。最后，讨论了J半交换性条件对某些类型的环的影响。