Nonunital prime rings graded by ordered groups

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.