关于正则环分离性的注记
Remarks on separativity of regular rings
摘要：
分离的冯诺依曼正则环大量存在。例如，所有的正则自内射环，单位正则环，具有多项式单位元的正则环都是分离的。是否存在非分离的正则环是一个未解问题。本文研究了vonNeumann正则环是分离环的各种条件。我们证明了von Neumann正则环$R$在下列任一情形下是分离的：{ \it（1）} $R$是CS; { \it（2）} $R$是伪内射的（自内射）; { \it（3）} $R$满足闭包扩展性质：两个同构右理想的$R$中的本质闭包是同构的。给出了正则透视环的另一个刻划（命题3.3）
英文摘要：
Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative regular ring. In this note, we study a variety of conditions under which a von Neumann regular ring is separative. We show that a von Neumann regular ring $R$ is separative under anyone of the following cases: {\it (1)} $R$ is CS; {\it (2)} $R$ is pseudo injective (auto-injective); {\it (3)} $R$ satisfies the closure extension property: the essential closures in $R$ of two isomorphic right ideals are themselves isomorphic. We also give another characterization of a regular perspective ring (Proposition 3.3)
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