On graded (1,r)-ideals

Abstract

Let G be a group with identity element e and R be a commutative G-graded ring with nonzero unity 1. In this paper, we introduce the graded version of (1,r)-ideals which is a generalization of graded r-ideals. A proper graded ideal I of R is said to be a graded (1,r)-ideal if whenever abc is an element of I for some nonunits homogeneous elements a,b,c is an element of h(R), then either ab is an element of I or c is an element of Z(G)(R). We investigate some basic properties of graded (1,r)-ideals. We show that if R admits a graded (1,r)-ideal that is not a graded r-ideal, then R is a G-graded local ring. Also, we give a method to construct graded (1,r)-ideals that are not graded r-ideals. Furthermore, we prove that R is a graded total quotient ring if and only if every proper graded ideal of R is graded (1,r)-ideal and also we present a counterpart of prime avoidance lemma for graded (1,r)-ideals. Finally, an idea is given about some graded (1,r)-ideals of the ring of fractions and the idealization.