Graded Strongly $r$-Ideals in a Graded Ring

Abstract

Let $R = \oplus_{\alpha \in G} R_{\alpha}$ be a commutative ring graded by any group $G$. This paper explores the concept of graded strongly $r$-ideals in graded rings, expanding upon the understanding of strongly $r$-ideals in ungraded contexts. It delves into the fundamental properties of these ideals, establishing relationships with graded $r$-ideals and graded $(1,r)$-ideals. The research further investigates these ideals in specific types of graded rings, including graded trivial ring extensions. Key results include methods for constructing graded $r$-ideals that are not graded strongly $r$-ideals, and characterizing these in the context of product rings.