Nonunital prime rings graded by ordered groups
Daniel Lännström, Patrik Lundström, Johan Öinert, Stefan Wagner
TL;DR
Let $G$ be an ordered group and $S$ a nonunital $G$-graded ring. The paper develops primeness criteria for graded rings in this nonunital setting, proving that for ordered $G$ the notions of prime and graded prime coincide, and establishing a correspondence between $G$-prime ideals of $S_e$ and graded prime ideals of $S$ under ideal-symmetric gradings. These results generalize the classical results of Nastăsescu and Van Oystaeyen to nonunital rings and yield practical criteria for Leavitt path rings and for prime graded subrings of group rings. The authors also show that a symmetrically graded subring can be prime without being nearly epsilon-strongly graded, and provide explicit constructions. Applications include a short, nonunital primeness criterion for Leavitt path rings and a general framework for primeness of symmetrically graded subrings of group rings.
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 Năstăsescu 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.