Some Generalizations of Green’s Relations in Rings and Modules

Abstract

In semigroups theory Green’s relations, introduced by J. Green, are a very important and useful tool for developing the semigroup theory. They characterise the element of a semigroup or a ring in terms of the principal ideals they generate. In contrast to early semigroup theory , where, as we have seen, ideas from ring’s were applied to semigroups, Green’s relation’s have also been applied to ring’s (Hollings, 2014). In ring theory Green’s relation’s are introduced by (Petro,2002) In this paper at first we generalize Green’s relations in rings. After this we notice that there exist an one to one correspondence between the ideals of a ring and this type of new relations we introduced.Then we compare them with Green’s relations in rings. At last we define some new relations in module theory, which mimic Green’s relations in rings, as an attempt to get tools in studying modules.