Cancellation properties of graded and nonunital rings. Graded clean and graded exchange Leavitt path algebras

TL;DR

This paper investigates how cancellation-related properties like clean and exchange behave in graded and nonunital rings, especially in Leavitt path algebras, and identifies graph conditions linked to these properties.

Contribution

It extends the concepts of clean and exchange rings to graded rings and explores their behavior in Leavitt path algebras, connecting graph properties to algebraic properties.

Findings

Characterization of graph properties equivalent to graded clean Leavitt path algebras

Identification of graph conditions necessary for graded exchange in Leavitt path algebras

Analysis of how graded matrix ring formation affects these properties

Abstract

Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and their "local" versions. We explore their relationships and extend the defined concepts to graded rings. With graded clean and graded exchange rings suitably defined, we study how these properties behave under the formation of graded matrix rings. We exhibit properties of a graph $E$ which are equivalent to the unital Leavitt path algebra $L_K(E)$ being graded clean. We also exhibit some graph properties which are necessary and some which are sufficient for $L_K(E)$ to be graded exchange.