具有自同态的强clean三角形矩阵环
Strongly clean triangular matrix rings with endomorphisms
摘要：
一个环$R$是强干净的，只要$R$中的每个元素都是幂等元与可交换单位之和。设$T_n(R,σ)$是局部环$R$上的斜三角形矩阵环，其中$σ$是$R$的自同态.我们证明了$T_2(R,σ)$是强干净的当且仅当对任何$a\in 1+J(R), b\in J(R)$，$l_a-r_{σ(b)}: R\to R$是满射的。此外，如果$l_{a}-r_{σ(b)}, l_{a}-r_{σ^2(b)}$和$l_{b}-r_{σ(a)}$对于任何$a\in U(R),b\in J(R)$都是满射，则$T_3(R,σ)$是强干净的。得到了$T_3(R,σ)$强洁净的必要条件。
英文摘要：
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,σ)$ be the skew triangular matrix ring over a local ring $R$ where $σ$ is an endomorphism of $R$. We show that $T_2(R,σ)$ is strongly clean if and only if for any $a\in 1+J(R), b\in J(R)$, $l_a-r_{σ(b)}: R\to R$ is surjective. Further, $T_3(R,σ)$ is strongly clean if $l_{a}-r_{σ(b)}, l_{a}-r_{σ^2(b)}$ and $l_{b}-r_{σ(a)}$ are surjective for any $a\in U(R),b\in J(R)$. The necessary condition for $T_3(R,σ)$ to be strongly clean is also obtained.
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