具有对合的自反环

Reflexive Rings with an Involution

本文研究了具有对合的环的自反性质.称环R的一个对合*是自反的,如果对任意a,b∈R,由aRb=0可推出bRa~*=0.若环R具有自反的对合*,则称R为*-自反环.我们对*-自反环的性质进行了刻画,并给出了一些具体的例子.作为应用,我们主要研究了与*-自反环相关的广义逆.对*-自反环R,我们证明了Moore-Penrose可逆元未必是群可逆元.

We study reflexive properties of rings with an involution.An involution * of a ring R is said to be reflexive if aRb=0 implies bRa~*=0 for all a,b ∈ R.A ring R with a reflexive involution * is called a *-reflexive ring.Some characterizations and examples of this class of rings are given.As applications,generalized inverses related to *-reflexive rings are studied.For a *-reflexive ring R,we show that a Moore-Penrose invertible element does not need to be group invertible.