弱$r$-Clean环

As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a∈R there exist an idempotent e and a regular element r such that a = r + e or a = r-e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean rings and weakly r-clean rings are equivalent for abelian rings.     

$(m,n)$-Igusa-Todorov 代数,IT-维数和三角矩阵代数

设$T$, $U$是两个Artin代数, $_U M_T$是$U$-$T$-双模.本文得到了三角矩阵代数$\Lambda=\left({\smallmatrix T&0\\ M&U \endsmallmatrix}\right)$是$(m,n)$-Igusa-Todorov的一个充要条件.我们还研究了$\Lambda$的IT维数.更具体地说,事实证明$$\max\{\ITdim T, \ITdim U\}\leqslant\ITdim \Lambda\leqslant\min\{\max\{\gldim T,\ITdim U\},\max\{\gldim U,\ITdim T\}\}.$$