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

2018/08/30 by Lännström, Daniel
#16P20 #16P40 #16S35 #16W50 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1808.10163

Abstract

Let G be a group with neutral element e and let S=\bigoplus_{g ∈ 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.

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