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 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.

Products of ideals in Leavitt path algebras

“…Proof. The equivalences (1) ⇔ (3) ⇔ (4) are established in [12,Theorem 5.7]. We shall show that (2) ⇒ (5) ⇒ (4) ⇒ (2).…”

“…has a nonzero constant term. According to [12,Lemma 5.4], gr(I) = gr(rad(I)) = gr(P ), and so gr(I) = I(H, S) is a prime ideal, by Theorem 2.8. Since necessarily gr(P ) = P , by the same theorem, We claim that f (x) = p n (x) for some positive integer n. Suppose, on the contrary, that there is an irreducible polynomial q( (5) ⇒ (4) This is a tautology.…”

“…Although Leavitt path algebras L are in general noncommutative, and as well possess plenty of zero divisors, recent investigations show that the (two-sided) ideals of such algebras share many properties with the ideals of commutative integral domains. For example, multiplication of ideals in L is commutative ( [1,Corollary 2.8.17] and [12,Theorem 3.4]); L is Bézout, that is all the finitely generated ideals of L are principal [11,Corollary 8]; L is hereditary (i.e., ideals are projective as L-modules), a property of Dedekind domains [3,Theorem 3.7]; and the ideal lattice of L is distributive [12,Theorem 4.3], a characterizing property of Prüfer domains among integral domains. (We note that the Bézout and hereditary properties hold for all one-sided ideals as well.)…”

“…Now suppose that I is not graded. Then, according to[12, Theorem 6.2], I is a product of prime ideals if and only if I(H, S) is the irredundant intersection of n > 0 prime ideals; andI/I(H, S) = k i=1 f i (c i ) , where k ≤ n, each c i is a cycle without exits in E \ (H, S), and f i (x) ∈ K[x]is a polynomial of smallest degree such that f i (c i ) ∈ I (which necessarily has a nonzero constant term). Now, by Theorem 2.3(2), in this situation I = I(H, S)+ k i=1 f i (c i ) if and only if I/I(H, S) = k i=1 f i (c i ) .…”

On Prüfer-like properties of Leavitt path algebras

“…Although a Leavitt path algebra L is non-commutative in nature and has plenty of zero divisors, it is somewhat intriguing and certainly interesting that the ideals of such a highly non-commutative algebra share many of the properties of the ideals of various types of (commutative) integral domains. To start with, the multiplication of ideals in L is commutative ( [1], [9,Theorem 3.4]), L satisfies the property of a Bézout domain, namely, all the finitely generated ideals of L are principal ( [8]), every ideal of L is projective, a property of Dedekind domains and the ideal lattice of L is distributive ( [9]) which characterizes Prüfer domains among integral domains.…”