The multiplicative ideal theory of Leavitt path algebras

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 distributive law A∩(B+C)=(A∩B)+(A∩C) holds for any three two-sided ideals of L. It is also shown that L is a multiplication ring, that is, given any two ideals A,B in L with A⊆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. The irreducible ideals of L turn out to be 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 that 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. Examples are constructed verifying whether some of the well-known theorems in the ideal theory of commutative rings such as the Cohen's theorem on prime ideals and the characterizing theorem on ZPI rings hold for Leavitt path algebras.