Commun. Math. Res., 34 (2018), pp. 241-252.
Published online: 2019-12
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Let $R$ be a ring and $J(R)$ the Jacobson radical. An element $a$ of $R$ is called (strongly) $J$-clean if there is an idempotent $e\in R$ and $w\in J(R)$ such that $a=e+w$ (and $ew=we$). The ring $R$ is called a (strongly) $J$-clean ring provided that every one of its elements is (strongly) $J$-clean. We discuss, in the present paper, some properties of $J$-clean rings and strongly $J$-clean rings. Moreover, we investigate $J$-cleanness and strongly $J$-cleanness of generalized matrix rings. Some known results are also extended.
}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2018.03.06}, url = {http://global-sci.org/intro/article_detail/cmr/13490.html} }Let $R$ be a ring and $J(R)$ the Jacobson radical. An element $a$ of $R$ is called (strongly) $J$-clean if there is an idempotent $e\in R$ and $w\in J(R)$ such that $a=e+w$ (and $ew=we$). The ring $R$ is called a (strongly) $J$-clean ring provided that every one of its elements is (strongly) $J$-clean. We discuss, in the present paper, some properties of $J$-clean rings and strongly $J$-clean rings. Moreover, we investigate $J$-cleanness and strongly $J$-cleanness of generalized matrix rings. Some known results are also extended.