The Multiplicative ideal theory of Leavitt Path Algebras

TL;DR

This paper explores the ideal structure of Leavitt path algebras, showing they are arithmetical and multiplication rings, and investigates their ideal factorizations, especially for finite graphs and certain algebraic conditions.

Contribution

It establishes that Leavitt path algebras are arithmetical and multiplication rings, and characterizes their ideal factorizations into prime, irreducible, or primary ideals.

Findings

Leavitt path algebras are arithmetical rings.

They are also multiplication rings, enabling ideal factorizations.

Ideals in finite graphs or under certain conditions factor into prime ideals.

Abstract

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the two-sided ideals of L form a distributive lattice. It is also shown that L is a multiplication ring, that is, given any two ideals A,B in L with A inside B, there is always an ideal C such that A = BC, an indication of a possible rich multiplicative ideal theory for L. Existence and uniqueness of factorization of the ideals of L as products of special types of ideals such as prime, irreducible or primary ideals is investigated. It turns out that the irreducible ideals of L are precisely the primary ideals of L. It is shown that an ideal I of L is a product of finitely many prime ideals if and only the graded part gr(I) of I is a product of prime ideals and I/gr(I) is finitely generated with a generating set of cardinality no more than the number of distinct prime ideals in the prime factorization of gr(I). As an application, it is shown that if E is a finite graph, then every ideal of L is a product of prime ideals. The same conclusion holds if L is two-sided artinian or two-sided noetherian.