On Strongly $J$-Semiclean Rings
Received:April 08, 2019  Revised:March 17, 2020
Key Word: upper triangular matrix   local ring   strongly $J$-semiclean ring
Fund ProjectL:Supported by the Excellent Course Project for Postgraduates of Hunan University of Science and Technology (Grant No.J52112).
 Author Name Affiliation Lunqun OUYANG Department of Mathematics, Hunan University of Science and Technology, Hunan 411201, P. R. China Zhaoqing GONG Department of Mathematics, Hunan University of Science and Technology, Hunan 411201, P. R. China
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We in this note introduce a new concept, so called strongly $J$-semiclean ring, that is a generalization of strongly $J$-clean rings. We first observe the basic properties of strongly $J$-semiclean rings, constructing typical examples. We next investigate conditions on a local ring $R$ that imply that the upper triangular matrix ring $T_n(R)$ is a strongly $J$-semiclean ring. Also, the criteria on strong $J$-semicleanness of $2\times 2$ matrices in terms of a quadratic equation are given. As a consequence, several known results relating to strongly $J$-clean rings are extended to a more general setting.