On intersections of two-sided ideals of Leavitt path algebras

TL;DR

This paper characterizes when all ideals in Leavitt path algebras are intersections of prime ideals, establishes uniqueness theorems for ideal representations, and explores properties of prime and graded ideals, including an analogue of Krull's theorem.

Contribution

It provides necessary and sufficient conditions for ideal intersections, introduces uniqueness theorems, and analyzes prime and graded ideals in Leavitt path algebras, extending classical ring theory results.

Findings

Ideals are intersections of prime ideals iff the graph satisfies Condition (K).

Unique representations of ideals as intersections or products of primes are established.

An analogue of Krull's theorem is proved for Leavitt path algebras.

Abstract

Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. Powers of a single ideal $I$ are considered and it is shown that the intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest graded ideal of $L$ contained in $I$. This leads to an analogue of Krull's theorem for Leavitt path algebras.