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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.
Acknowledgments
Notes
A quasi-ideal in a ring is said to have the intersection property if it is the intersection of a left ideal and a right ideal., then the relations and
coincide in
.
In view of the duality between and
it suffices to prove the theorem for relation
in place of
.