环中幂等元的交换子与反交换子
Commutators and Anti-Commutators of Idempotents in Rings
摘要：
证明了环$\,R\,$有两个幂等元$\,e,e'\,$和一个可逆交换子$\,ee'-e'e\,$当且仅当对环$\,S\,$有$\,R \cong {\mathbb M}_2(S)\,$，其中$\,1\,$是两个单位之和.在这种情况下，“反交换子”$\,ee'+e'e\,$是自动可逆的，所以我们也研究了更广泛的一类环具有这样的可逆反交换子。具有上述性质之一的简单Artin环$\,R\,$（以及其他相关的矩阵环类）被完全确定。在本研究中，我们还得到了{ \it一般\} $\,2\times 2\,$矩阵环的各种新的判别准则。例如，$R\,$是这样的矩阵环当且仅当它有可逆交换子$\,er-re\,$，其中$\,e^2=e$。
英文摘要：
We show that a ring $\,R\,$ has two idempotents $\,e,e'\,$ with an invertible commutator $\,ee'-e'e\,$ if and only if $\,R \cong {\mathbb M}_2(S)\,$ for a ring $\,S\,$ in which $\,1\,$ is a sum of two units. In this case, the "anti-commutator" $\,ee'+e'e\,$ is automatically invertible, so we study also the broader class of rings having such an invertible anti-commutator. Simple artinian rings $\,R\,$ (along with other related classes of matrix rings) with one of the above properties are completely determined. In this study, we also arrive at various new criteria for {\it general\} $\,2\times 2\,$ matrix rings. For instance, $R\,$ is such a matrix ring if and only if it has an invertible commutator $\,er-re\,$ where $\,e^2=e$.
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