# Two-Sided Ideals in Leavitt Path Algebras

## TL;DR

This paper characterizes two-sided ideals in Leavitt path algebras for row-finite graphs, showing they are generated by specific cycle-based elements, and links algebraic properties to graph-theoretic conditions.

## Contribution

It provides an explicit description of two-sided ideals in Leavitt path algebras and establishes a criterion for the algebra being two-sided Noetherian based on graph properties.

## Key findings

- Two-sided ideals are generated by elements involving cycles at vertices.
- Leavitt path algebra is two-sided Noetherian iff certain chain conditions hold for graph subsets.
- The description connects algebraic properties with graph-theoretic conditions.

## Abstract

We explicitly describe two-sided ideals in Leavitt path algebras associated with a row-finite graph. Our main result is that any two-sided ideal $I$ of a Leavitt path algebra associated with a row-finite graph is generated by elements of the form $v + \sum_{i=1}^n\lambda_i g^i$, where $g$ is a cycle based at vertex $v$. We use this result to show that a Leavitt path algebra is two-sided Noetherian if and only if the ascending chain condition holds for hereditary and saturated closures of the subsets of the vertices of the row-finite graph $E$.