Generalizations of Graded $S$-Primary Ideals

TL;DR

This paper introduces and explores new classes of graded weakly $S$-primary ideals in commutative graded rings, extending the concept of graded weakly primary ideals with detailed properties and characteristics.

Contribution

It defines graded weakly $S$-primary and $g$-weakly $S$-primary ideals, expanding the theory of graded primary ideals with new extensions and properties.

Findings

Characterization of graded weakly $S$-primary ideals

Properties of graded $g$-weakly $S$-primary ideals

Conditions under which these ideals exist and their behavior

Abstract

The goal of this article is to present the graded weakly $S$-primary ideals and $g$-weakly $S$-primary ideals which are extensions of graded weakly primary ideals. Let $R$ be a commutative graded ring, $S\subseteq h(R)$ and $P$ be a graded ideal of $R$. We state $P$ is a graded weakly $S$-primary ideal of $R$ if there exists $s\in S$ such that for all $x,y \in h(R)$, if $0\neq xy\in P$, then $sx\in P$ or $sy\in Grad(P)$ (the graded radical of $P$). Several properties and characteristics of graded weakly $S$-primary ideals as well as graded $g$-weakly $S$-primary ideals are investigated.