Graded quasi-S-primary ideals

Abstract

Let [Formula: see text] be a group, [Formula: see text] a [Formula: see text]-graded commutative ring with nonzero unity [Formula: see text] and [Formula: see text] a multiplicative subset of [Formula: see text]. This paper introduces the concept of graded quasi-[Formula: see text]-primary (respectively, graded weakly quasi-[Formula: see text]-primary) ideals. A properly graded ideal [Formula: see text] of [Formula: see text] is said to be graded quasi-[Formula: see text]-primary (respectively, graded weakly quasi-[Formula: see text]-primary) if whenever the nonunit elements [Formula: see text] such that [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] or [Formula: see text] for some [Formula: see text] with [Formula: see text] being the graded radical of [Formula: see text]. We investigate some basic properties of graded quasi-[Formula: see text]-primary (respectively, graded weakly quasi-[Formula: see text]-primary) ideals. We discuss the form of graded quasi-[Formula: see text]-primary (respectively, graded weakly quasi-[Formula: see text]-primary) ideals in a finite direct product. Furthermore, we study graded quasi-[Formula: see text]-primary (respectively, graded weakly quasi-[Formula: see text]-primary) ideals in Nagata’s idealization ring.