Abstract
ABSTRACT In this paper we study the Green's relations , , , , as well as the relations concerning principal quasi-ideals in rings, whose definitions mimic definitions of Green's relations and relation in semigroups. We show that, differently from semigroups in rings we have in general , and we provide a sufficient, but not necessary, condition for , to hold in a ring. We prove some results about Green's relation in rings, which are similar to the analogous result in semigroups. The main result is an analogous of Green's Theorem for semigroups. By means of Green's Theorem for rings we give a short proof of Theorem about minimal quasi-ideal in rings. We provide various sufficient, but not necessary conditions on a minimal quasi-ideal of a ring for the set to be a -class. Finally we prove that, if for some elements of a ring we have and the principal quasi-ideal is minimal and has the intersection property, then the principal quasi-ideal is minimal and has the intersection property, too.
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