On Graded (1,r)-Ideals

Abstract

Let [Formula: see text] be a group with identity element [Formula: see text] and [Formula: see text] be a commutative [Formula: see text]-graded ring with nonzero unity [Formula: see text]. In this paper, we introduce the graded version of [Formula: see text]-ideals which is a generalization of graded [Formula: see text]-ideals. A proper graded ideal [Formula: see text] of [Formula: see text] is said to be a graded [Formula: see text]-ideal if whenever [Formula: see text] for some nonunits homogeneous elements [Formula: see text], then either [Formula: see text] or [Formula: see text]. We investigate some basic properties of graded [Formula: see text]-ideals. We show that if [Formula: see text] admits a graded [Formula: see text]-ideal that is not a graded [Formula: see text]-ideal, then [Formula: see text] is a [Formula: see text]-graded local ring. Also, we give a method to construct graded [Formula: see text]-ideals that are not graded [Formula: see text]-ideals. Furthermore, we prove that [Formula: see text] is a graded total quotient ring if and only if every proper graded ideal of [Formula: see text] is graded [Formula: see text]-ideal and also we present a counterpart of prime avoidance lemma for graded [Formula: see text]-ideals. Finally, an idea is given about some graded [Formula: see text]-ideals of the ring of fractions and the idealization.