No abstract

In this paper we show that the identity function e of a generalized ring R is a generalized ring homomorphism if e(x + y) = e(x) + e(y), for all x, y e R. Moreover, we show that if R is a generalized ring with an identity, then e is a generalized ring homomorphism. Properties of identities and identity mappings of generalized rings are considered. A method for constructing a new generalized ring with an identity via a given quotient generalized ring with an identity, is presented. Second isomorphism theorem and third isomorphism theorem for M-rings are proved.

The aim of this paper is to consider the ringswhich can be graded by completely simple semigroups. We show that each G-graded ring has an orthonormal basis, where G is a completely simple semigroup. We prove that if I is a complete homogeneous ideal of a G-graded ring R, then R/I is a G-graded ring.We deduce a characterization of the maximal ideals of a G-graded ring which are homogeneous. We also prove that if R is a Noetherian graded ring, then each summand of it is also a Noetherian module..