Leavitt path algebras are B\'ezout

Gene Abrams; Francesca Mantese; and Alberto Tonolo

arXiv:1605.08317·math.RA·May 27, 2016

Leavitt path algebras are B\'ezout

Gene Abrams, Francesca Mantese, and Alberto Tonolo

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TL;DR

This paper proves that Leavitt path algebras over any field are Be9zout rings, meaning every finitely generated one-sided ideal is principal, which advances understanding of their algebraic structure.

Contribution

It establishes that all Leavitt path algebras are Be9zout rings, a new structural property for these algebras.

Findings

01

Leavitt path algebras are Be9zout rings.

02

Every finitely generated one-sided ideal in these algebras is principal.

Abstract

Let $E$ be a directed graph, $K$ any field, and let $L_{K} (E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$ . We show that $L_{K} (E)$ is a B\'{e}zout ring, i.e., that every finitely generated one-sided ideal of $L_{K} (E)$ is principal.

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