强$J$-Semiclean环

On Strongly $J$-Semiclean Rings

中文关键词:  上三角形  局部环  强$J$-semiclean环

英文关键词:upper triangular matrix  local ring  strongly $J$-semiclean ring

中文摘要:

      作为强$J$-clean环的推广,我们在这篇文章中引进了强$J$-semiclean环的定义. 我们首先通过构造大量典型例子探讨了强$J$-semiclean环的基本性质;其次我们得到了局部环$R$上$n$阶上三角形矩阵环$T_n(R)$是强$J$-semiclean环时局部环$R$应满足的条件; 同时我们得到了如何通过某些特殊的二次方程来判断环上二阶方阵是否是强$J$-semiclean环的方法.在这篇文章中,我们把强$J$-clean环的一些结论推广到了更一般的情形.

英文摘要:

      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.