Chain conditions for epsilon-strongly graded rings with applications to Leavitt path algebras

TL;DR

This paper extends the understanding of noetherian and artinian properties in epsilon-strongly graded rings, especially applied to Leavitt path algebras, broadening previous results to more general coefficient rings.

Contribution

It generalizes noetherianity and artinianity criteria for epsilon-strongly graded rings and applies these results to characterize Leavitt path algebras over arbitrary unital rings.

Findings

Epsilon-strongly graded rings' noetherianity depends on the principal component.

Characterizations of noetherian and artinian Leavitt path algebras over general unital rings.

Extension of previous results from fields and commutative rings to broader coefficient rings.

Abstract

Let $G$ be a group with neutral element $e$ and let $S=\bigoplus_{g \in G}S_g$ be a $G$-graded ring. A necessary condition for $S$ to be noetherian is that the principal component $S_e$ is noetherian. The following partial converse is well-known: If $S$ is strongly-graded and $G$ is a polycyclic-by-finite group, then $S_e$ being noetherian implies that $S$ is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products.