Nonunital prime rings graded by ordered groups

TL;DR

This paper explores the structure of nonunital graded rings, establishing conditions under which prime ideals and primeness of the entire ring relate, especially for ordered groups, extending classical results to broader contexts.

Contribution

It generalizes classical theorems on prime rings to nonunital, graded settings, especially for ordered groups, and introduces new correspondences between graded and ungraded prime ideals.

Findings

Graded prime ideals coincide with prime ideals in ordered group settings.

Established a bijective correspondence between graded prime ideals of the ring and prime ideals of the degree-zero component.

Provided new criteria for primeness of Leavitt path rings and symmetrically graded subrings.

Abstract

Let $G$ be a group with identity element $e$, and suppose that $S$ is an associative $G$-graded ring that is not necessarily unital. In the case where $G$ is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group $G$, if $S$ is what we call ideally symmetrically $G$-graded, then we show that there is a bijective correspondence between the $G$-graded prime ideals of $S$ and the $G$-prime ideals of $S_e$. We use this correspondence in the case where $G$ is ordered and $S$ is ideally symmetrically $G$-graded to show that $S$ is prime if and only if $S_e$ is $G$-prime. These results generalize classical theorems by N\u{a}st\u{a}sescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically $G$-graded subrings of group rings over fully idempotent rings.