Every finitely generated ideal of a Leavitt path algebra is a principal ideal

TL;DR

This paper proves that in Leavitt path algebras over any graph and field, all finitely generated ideals are principal, extending to all ideals in finite graphs, thus clarifying their ideal structure.

Contribution

It provides an explicit description of generators for non-graded ideals and establishes that all finitely generated ideals are principal, including in finite graphs.

Findings

Every finitely generated ideal in L_{K}(E) is principal.

In finite graphs, all ideals are principal.

Explicit generators for non-graded ideals are given.

Abstract

Let E be an arbitrary graph and K be any field. For every non-graded ideal I of the Leavitt path algebra L_{K}(E), we give an explicit description of the generators of I. Using this, we show that every finitely generated ideal of L_{K}(E) must be principal. In particular, if E is a finite graph, then every ideal of L_{K}(E) must be principal ideal.