# Largest ideals in Leavitt path algebras

## TL;DR

This paper characterizes the largest ideals in Leavitt path algebras, including those that are semisimple, noetherian, exchange, and purely infinite, and studies their invariance under ring isomorphisms.

## Contribution

It provides a comprehensive description of the largest ideals in Leavitt path algebras and analyzes their invariance properties under ring isomorphisms.

## Key findings

- Largest locally left/right artinian ideal identified
- Largest locally left/right noetherian ideal characterized
- Largest purely infinite ideal described as a direct sum

## Abstract

We identify largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely infinite non-simple and non-decomposable pieces. The invariance under ring isomorphisms of these ideals is also studied.