Perspective rings

In this paper, we study the modules whose any two isomorphic summands have a common complement. We call such modules perspective  . It is proved that perspectivity is an ER-property, in the sense that it depends only on the endomorphism ring of the module. This property in rings turns out to be left–right symmetric, that is, RRRR is perspective if and only if RR is perspective for any ring R and we call such ring a perspective ring. The class of perspective rings lies properly between the class of rings with stable range one and the class of rings with the internal cancellation property. We give several characterizations of perspective rings and also study perspectivity in related rings. It is proved that R   has stable range one if and only if M2(R)M2(R) is a perspective ring. Another characterization of rings with stable range one, in terms of completions of unimodular rows, is also given. We prove that every regular element in a perspective ring is clean.