Graded $n$-semiprimary ideals in a graded ring

Abstract

Let $R=\oplus_{\gamma \in \Gamma} R_{\gamma}$ be a commutative ring graded by a commutative monoid $\Gamma$, and let $n$ be a positive integer. This paper introduces the concept of graded $n$-semiprimary ideals in graded rings, extending the framework of $n$-semiprimary ideals beyond ungraded structures. We establish foundational properties of these ideals, distinguishing them from their ungraded counterparts and analyzing their connections to graded semiprimary ideals. Additionally, we examine graded $n$-semiprimary ideals in specific graded rings, such as graded trivial ring extensions and graded amalgamations. Key results include construction techniques for graded $n$-semiprimary ideals and characterizations in product rings, offering comprehensive insights into their structural properties and behaviors in graded settings.