On simple modules over Leavitt path algebras

Journal of Algebra 423 (2015) 239–258

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Journal of Algebra www.elsevier.com/locate/jalgebra

On simple modules over Leavitt path algebras Kulumani M. Rangaswamy University of Colorado at Colorado Springs, Colorado Springs, CO 80918, United States

a r t i c l e

i n f o

Article history: Received 15 February 2014 Available online 24 October 2014 Communicated by Louis Rowen Dedicated to Professor Laszlo Fuchs on the occasion of his 90th birthday MSC: 16D70 Keywords: Leavitt path algebras Arbitrary graphs Simple modules Primitive ideals Finitely presented simple modules

a b s t r a c t Given an arbitrary graph E and any field K, a new class of simple modules over the Leavitt path algebra LK (E) is constructed by using vertices that emit infinitely many edges in E. The corresponding annihilating primitive ideals are also described. Given a fixed simple LK (E)-module S, we compute the cardinality of the set of all simple LK (E)-modules isomorphic to S. Using a Boolean subring of idempotents induced by paths in E, bounds for the cardinality of the set of distinct isomorphism classes of simple LK (E)-modules are given. We also obtain a complete structure theorem about the Leavitt path algebra LK (E) of a finite graph E over which every simple module is finitely presented. © 2014 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries Leavitt path algebras were introduced in [1,8] as algebraic analogues of graph C*-algebras and as natural generalizations of Leavitt algebras of type (1, n) built in [18]. The various ring-theoretical properties of these algebras have been actively investigated in a series of papers (see, e.g., [1,3,5,8,13,19,20,24]). In contrast, the module theory of Leavitt path algebras LK (E) of arbitrary directed graphs E over a field K is still at its

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2014.10.008 0021-8693/© 2014 Elsevier Inc. All rights reserved.

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infancy. The initial organized attempt to study LK (E)-modules was done in [7] where, for a finite graph E, the simply presented LK (E)-modules were described in terms of ¯ of E. finite dimensional representations of the usual path algebras of the reverse graph E As an important step in the study of modules over a Leavitt path algebra LK (E), the investigation of the simple LK (E)-modules has recently received some attention (see [10–12,14]). Following the ideas of Smith [22], Chen [14] constructed irreducible representations of LK (E) by using sinks and tail-equivalent classes of infinite paths in the graph E. Chen’s construction was expanded in [10] to introduce additional classes of non-isomorphic simple LK (E)-modules. In Section 2 of this paper, we construct a new class of simple left LK (E)-modules induced by vertices which are infinite emitters and at the same time streamline the process of construction of certain simple modules introduced in [10]. By describing the annihilating primitive ideals of these simple modules, we show that these new simple modules are distinct from (i.e., not isomorphic to) any of the previously constructed simple LK (E)-modules in [10,12,14]. In Section 3, we adapt the ideas of Rosenberg [21] and compute the cardinality of a single isomorphism class of simple left LK (E)-modules isomorphic to a given simple LK (E)-module (Theorem 3.5). Using a Boolean ring consisting of idempotents induced by the paths in E, we also obtain a lower bound for the size of the set of distinct isomorphism classes of simple LK (E)-modules. In particular, if LK (E) is a countable dimensional simple algebra, then it will have exactly 1 or at least 2ℵ0 distinct isomorphism classes of simple modules. In Section 4, we consider Leavitt path algebras over which every simple module is finitely presented. In [10] it was shown that, given a finite graph E, all the simple modules over LK (E) are finitely presented exactly when every vertex in E is the base of at most one cycle. Interestingly, it was shown in [5] that this same property on the finite graph E is equivalent to the Leavitt path algebra LK (E) having finite Gelfand–Kirillov dimension. In addition, it was shown in [6] that if E has this property, then necessarily LK (E) is the union of a finite ascending chain of ideals 0 ⊂ I0 ⊂ I1 ⊂ · · · ⊂ Im = L

(++)

where I0 is a direct sum of finitely many matrix rings Mn (K) with n ∈ N ∪ {∞} and for each j ≥ 0, Ij+1 /Ij is a direct sum of finitely many matrix rings Mni (K) over the field K and finitely many matrix rings Mli (K[x, x−1 ]) over the Laurent polynomial ring K[x, x−1 ] with ni , li ∈ N ∪ {∞}. We first strengthen the above necessary condition by showing that, if E has the stated property, then in addition to the above condition on the ideal I0 in the chain (++), we have, for each j ≥ 0, (i) Ij is a graded ideal, (ii) Soc(L/Ij ) = 0, (iii) the ideal Ij+1 /Ij is a finite direct sum of matrix rings over K[x, x−1 ] only, say Mli K[x, x−1 ] with li ∈ N ∪ {∞} (and does not involve matrix rings over K) and (iv) Ij+1 /Ij contains every other ideal N/Ij of L/Ij which is a direct sum of matrix rings over K[x, x−1 ]. We then show that the converse of this strengthened version

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of the statement of Theorem 1 of [6] holds. This leads to a complete ideal-theoretic structural characterization of the Leavitt path algebras L of a finite graph E over which every simple module is finitely presented (see Theorem 4.3 below). Moreover, the extensions Ij+1 /Ij do not split for all j ≥ 1, thus answering a remark mentioned in [6]. In passing, an easy proof of Theorem 2 of [6] is also pointed out. For the general notation, terminology and results in Leavitt path algebras, we refer to [1,2,8]. We give below a short outline of some of the needed basic concepts and results. A (directed) graph E = (E 0 , E 1 , r, s) consists of two sets E 0 and E 1 together with maps r, s : E 1 → E 0 . The elements of E 0 are called vertices and the elements of E 1 edges. All the graphs E that we consider (excepting those studied in Section 4) are arbitrary in the sense that no restriction is placed either on the number of vertices in E or on the number of edges emitted by a single vertex. Also K stands for an arbitrary field. A vertex v is called a sink if it emits no edges and a vertex v is called a regular vertex if it emits a non-empty finite set of edges. An infinite emitter is a vertex which emits infinitely many edges. For each e ∈ E 1 , we call e∗ a ghost edge. We let r(e∗ ) denote s(e), and we let s(e∗ ) denote r(e). A path μ of length n > 0 is a finite sequence of edges μ = e1 e2 · · · en with r(ei ) = s(ei+1 ) for all i = 1, · · ·, n − 1. In this case μ∗ = e∗n · · · e∗2 e∗1 is the corresponding ghost path. A vertex is considered a path of length 0. The set of all vertices on the path μ is denoted by μ0 . A path μ = e1 . . . en in E is closed if r(en ) = s(e1 ), in which case μ is said to be based at the vertex s(e1 ). A closed path μ as above is called simple provided it does not pass through its base more than once, i.e., s(ei ) = s(e1 ) for all i = 2, ..., n. The closed path μ is called a cycle if it does not pass through any of its vertices twice, that is, if s(ei ) = s(ej ) for every i = j. If there is a path from vertex u to a vertex v, we write u ≥ v. A subset D of vertices is said to be downward directed if for any u, v ∈ D, there exists a w ∈ D such that u ≥ w and v ≥ w. A subset H of E 0 is called hereditary if, whenever v ∈ H and w ∈ E 0 satisfy v ≥ w, then w ∈ H. A hereditary set is saturated if, for any regular vertex v, r(s−1 (v)) ⊆ H implies v ∈ H. Given an arbitrary graph E and a field K, the Leavitt path algebra LK (E) is defined to be the K-algebra generated by a set {v : v ∈ E 0 } of pairwise orthogonal idempotents together with a set of variables {e, e∗ : e ∈ E 1 } which satisfy the following conditions: (1) (2) (3) (4)

s(e)e = e = er(e) for all e ∈ E 1 . r(e)e∗ = e∗ = e∗ s(e) for all e ∈ E 1 . (The “CK-1 relations”) For all e, f ∈ E 1 , e∗ e = r(e) and e∗ f = 0 if e = f . (The “CK-2 relations”) For every regular vertex v ∈ E 0 ,  v= ee∗ . e∈E 1 ,s(e)=v

For any vertex v, the tree of v is T (v) = {w : v ≥ w}. We say there is a bifurcation at a vertex v, if v emits more than one edge. In a graph E, a vertex v is called a line

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point if there is no bifurcation or a cycle based at any vertex in T (v). Thus, if v is a line point, there will be a single finite or infinite line segment μ starting at v (μ could just be v) and any other path α with s(α) = v will just be an initial subpath of μ. It was shown in [12] that v is a line point in E if and only if LK (E)v (and likewise vLK (E)) is a simple left (right) ideal. We shall be using the following concepts and results from [24]. A breaking vertex of a hereditary saturated subset H is an infinite emitter w ∈ E 0 \H with the property that 0 < |s−1 (w) ∩ r−1 (E 0 \H)| < ∞. The set of all breaking vertices of H is denoted by BH .  ∗ For any v ∈ BH , v H denotes the element v − s(e)=v,r(e)∈H / ee . Given a hereditary saturated subset H and a subset S ⊆ BH , (H, S) is called an admissible pair. Given an admissible pair (H, S), the ideal generated by H ∪ {v H : v ∈ S} is denoted by I(H,S) . It was shown in [24] that the graded ideals of LK (E) are precisely the ideals of the form I(H, S) for some admissible pair (H, S). Moreover, LK (E)/I(H, S) ∼ = LK (E\(H, S)). 0 Here E\(H, S) is the Quotient graph of E in which (E\(H, S)) = (E 0 \H) ∪ {v  : v ∈ / H} ∪ {e : e ∈ E 1 , r(e) ∈ BH \S} and r, s BH \S} and (E\(H, S))1 = {e ∈ E 1 : r(e) ∈ 0 are extended to (E\(H, S)) by setting s(e ) = s(e) and r(e ) = r(e) . In particular, if E is a finite graph and H is a hereditary saturated set of vertices, BH is empty and we define the Quotient graph E\H corresponding to H by taking BH and S in the above definition to be empty sets. A useful observation is that every element a of LK (E) can be written as a = n ∗ ki ∈ K, αi , βi are paths in E and n is a suitable integer. Morei=1 ki αi βi , where   over, LK (E) = v∈E 0 LK (E)v = v∈E 0 vLK (E) (see [1]). Even though the Leavitt path algebra LK (E) may not have the multiplicative identity 1, we shall write LK (E)(1 − v) to denote the set {x − xv : x ∈ LK (E)}. If v is an idempotent (in particular, a vertex), we get a direct decomposition LK (E) = LK (E)v ⊕ LK (E)(1 − v). In Section 4 we shall use the notation M∞ (R) to denote the ring of ω × ω infinite matrices over R with only finitely many non-zero entries from a ring R. 2. A new class of simple modules Let E be an arbitrary graph. For convenience in writing, we shall denote the Leavitt path algebra LK (E) by L. Throughout this section, we shall use the following notation. For v ∈ E 0 define     M (v) = w ∈ E 0 : w ≥ v and H(v) = E 0 \ M (v) = u ∈ E 0 : u  v . Clearly M (v) is downward directed. Also, for any vertex v which is a sink or infinite emitter, the set H(v) is a hereditary saturated subset of E. If v is a finite emitter, it might be that H(v) is not saturated, and that v belongs to the saturation of H(v). Definition 2.1 (The L-module Sv∞ ). Suppose v is an infinite emitter in E. Define Sv∞ to be the K-vector space having as a basis the set B = {p : p a path in E with r(p) = v}.

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Following Chen [14], we define, for each vertex u and each edge e in E, linear transformations Pu , Se and Se∗ on Sv∞ as follows: For all paths p ∈ B,  Pu (p) =  Se (p) =

p, 0,

if u = s(p) otherwise

ep, if r(e) = s(p) 0, otherwise

Se∗ (v) = 0   p , if p = ep Se∗ (p) = 0, otherwise Then it is straightforward to check that the endomorphisms {Pu , Se , Se∗ : u ∈ E 0 , e ∈ E 1 } satisfy the defining relations (1)–(4) of the Leavitt path algebra L. This induces an algebra homomorphism φ from L to End K (Sv∞ ) mapping u to Pu , e to Se and e∗ to Se∗ . Then Sv∞ can be made a left module over L via the homomorphism φ. We denote this L-module operation on Sv∞ by ·. Remark 2.2. The above construction does not work if v is a regular vertex. Specifically,  the needed CK-2 relation e∈s−1 (v) Se Se∗ = Pv does not hold. Because, on the one hand  ( e∈s−1 (v) Se Se∗ )(v) = 0 but on the other hand Pv (v) = v = 0. Proposition 2.3. For each infinite emitter v in E, Sv∞ is a simple left L-module. Proof. Suppose U is non-zero submodule of Sv∞ . Let 0 = a =

n 

ki pi ∈ U

(#)

i=1

where ki ∈ K and the pi are paths in E with r(pi ) = v and we assume that the paths pi are all different. By induction on the number of terms n in (#), we wish to show that v ∈ U . Suppose n = 1 so that a = k1 p1 . Then p∗1 · a = k1 v ∈ U and we are done. Suppose n > 1 and assume that v ∈ U if U contains a non-zero element which is a K-linear combination of less than n paths. Among the paths pi in Eq. (#), assume that p1 has the smallest length. If p1 has length 0, that is, if p1 = v and, for some s, ps is a path of length > 0, then since p∗s · v = 0, 0 = p∗s · a ∈ U will be a sum of less than n terms and so, by induction, v ∈ U . Suppose p1 has length > 0. Now p∗1 · a = a ∈ U . If p∗1 · a is a sum of less than n terms, we are done. Otherwise, a = k1 p∗1 · p1 + b = k1 v + b where b is a sum of less than n terms with its first non-zero term, say, kt pt . Then p∗t · a = p∗t · b ∈ U and 0 = p∗t · b is a sum of less than n terms. Hence by induction, v ∈ U and we conclude that U = Sv∞ . 2

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Proposition 2.5 below describes the annihilating primitive ideal of the simple module Sv∞ . Crucial to its proof is the following theorem (Theorem 4.2, [19]) describing the structure primitives ideals in an arbitrary Leavitt path algebra. Theorem 2.4. (See [19].) Let E be an arbitrary graph. If P is a primitive ideal of L with P ∩ E 0 = H, then P satisfies one of the following properties: (i) P is a non-graded ideal and is the ideal generated by I(H, BH ) ∪ {f (c)}, where c is a cycle without exits in E 0 \H based at a vertex v and E 0 \H = M (v); (ii) P is a graded ideal of the form P = I(H, BH \{u}) where u ∈ BH and E 0 \H = M (u); (iii) P is a graded ideal of the form P = I(H, BH ), E 0 \H is downward directed, satisfies Condition (L) and the countable separation property. Proposition 2.5. Let v be an infinite emitter in the graph E. Then ⎧ I(H(v), BH(v) ), ⎪ ⎨ Ann L (Sv∞ ) = I(H(v), BH(v) \{v}), ⎪ ⎩ I(H(v), BH(v) ),

if |s−1 (v) ∩ r−1 (M (v))| = 0; if 0 < |s−1 (v) ∩ r−1 (M (v))| < ∞ if |s−1 (v) ∩ r−1 (M (v))| is infinite.

Proof. Let J = Ann L (Sv∞ ). Clearly H(v) ⊂ J since for any u ∈ H(v), u  v and so u · p = 0 for all p with r(p) = v. Indeed J ∩ E 0 = H(v). Suppose |s−1 (v) ∩ r−1 (M (v))| = 0 so that r(s−1 (v))  H(v). Let u ∈ BH(v) . Clearly H(v) u · p = 0 if p is a path with r(p) = v and s(p) = u. On the other hand, if p is a path from u to v with p = ep where e is an edge, then u

H(v)

·p=

u−



ff

∗

· p = (e − e)p = 0.

f ∈s−1 (u),r(f )∈H(v) /

This shows that I(H(v), BH(v) ) ⊆ J. Now J ∩ E 0 = H(v). We claim that the primitive ideal J is a graded ideal. Because, otherwise it follows Theorem 2.4(i), that v will be the base of a cycle c with c0 ⊂ M (v). In particular, there is an edge e with s(e) = v and r(e) ∈ M (v). But this is not possible since r(s−1 (v)) ⊆ H(v). Thus J is a graded ideal with J ∩ E 0 = H(v). Since I(H(v), BH(v) ) is the largest graded ideal for which I(H(v), BH(v) ) ∩ E 0 = H(v), we conclude that J = I(H(v), BH(v) ). Suppose 0 < |s−1 (v) ∩ r−1 (M (v))| < ∞. This means v ∈ BH(v) . If u ∈ BH(v) with u = v, then the arguments in the preceding paragraph shows that uH(v) ∈ J. On the other hand, v H(v) ∈ / J since v

H(v)

·v =

v−

 f ∈s−1 (v),L(f )∈H(v) /

ff

∗

· v = v · v − 0 = v = 0.

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This shows that I(H(v), BH(v) \{v}) ⊆ J. Now J is a primitive ideal with J ∩E 0 = H(v). As before, we claim J is a graded ideal. Because, otherwise from Theorem 2.4(i), we will have I(H(v), BH(v) )  J and this is not possible since v H(v) ∈ / J. We then conclude that J is a graded ideal and that J = I(H(v), BH(v) \{v}). Finally, suppose |s−1 (v) ∩ r−1 (M (v))| is infinite. This means, in particular, there are infinitely many closed paths in M (v) based at v. As M (v) = E 0 \H(v), it is then clear from Theorem 2.4(i) that J cannot be a non-graded ideal. Also, as v ∈ / BH(v) , the earlier arguments show that I(H(v), BH(v) ) ⊆ J. Observing that I(H(v), BH(v) ) ∩E 0 = H(v) = J ∩ E 0 , we then conclude that J = I(H(v), BH(v) ). 2 Before proceeding further, we shall review the construction of some of the simple left L-modules introduced in [14] and [10]. In this connection, we wish to point out that the notation and terminology used by Chen in [14] is different from those normally used in papers on Leavitt path algebras such as [2,10]. We shall follow the notation of [2] when describing the simple modules defined by Chen. A) The Chen Simple Module V[p] : Chen [14] defines an equivalence relation among infinite paths in E by using the following notation. If p = e1 e2 · · · en · · · is an infinite path where the ei are edges, then for any positive integer n, let τ≤n (p) = e1 e2 · · · en and τ>n (p) = en+1 en+2 · · · . Two infinite paths p and q are said to be tail equivalent, in symbols, p ∼ q, if there exist positive integers m and n such that τ>n (p) = τ>m (q). Then  is an equivalence relation. Given an equivalence class of infinite paths [p], let V[p] denote the K-vector space having the set {q : q ∈ [p]} as a basis. Then Chen [14] defines an L-module operation on V[p] making V[p] a left L-module similar to the way the module operation is defined on Sv∞ above, except that the condition that Se∗ (v) = 0 is dropped for any edge e. Chen [14] shows that the module V[p] becomes a simple left L-module. B) The Chen Simple Module Nw : Let w be a sink and Nw be a K-vector space having as a basis the set {p : p paths in E with r(p) = w}. Proceeding as was done above in the definition of Sv∞ , Chen [14] defines an L-module action on Nw and shows that Nw becomes a simple module. C) The three classes of simple L-modules introduced in [10], denoted respectively by B H(v) f Nv H(v) , Nv and V[p] : B

(i) The Simple Module Nv H(v) : Suppose that v is an infinite emitter such that v ∈ BH(v) . Then we can build the primitive ideal P = I(H(v), BH(v) \ {v}) (see [19]) and the factor ring LK (E)/P ∼ = LK (F ) where F = E \ (H(v), BH(v) \ {v}). Here F 0 = (E 0 \ H(v)) ∪ {v  },     F 1 = e ∈ E 1 : r(e) ∈ / H(v) ∪ e : e ∈ E 1 , r(e) = v

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and r and s are extended to F by s(e ) = s(e) and r(e ) = v  for all e ∈ E 1 with r(e) = v. Note that v  is a sink in F and it is easy to see that MF (v  ) = F 0 . Accordingly, we may consider the Chen simple module Nv of LK (F ) corresponding to the sink v  in F as defined in paragraph B) above. Using the quotient map LK (E) → LK (F ), we may view Nv as a simple module over L = LK (E). This simple L-module B is denoted by Nv H(v) . H(v) (ii) The Simple Module Nv : Suppose v is an infinite emitter and such that r(s−1 (v)) ⊆ H(v). Then v is the unique sink in the graph G = E \ (H(v), BH(v) ). Let Nv be the Chen simple LK (E \(H(v), BH(v) ))-module corresponding to the sink v. It is clear that Nv is a faithful simple LK (G)-module. Consider Nv as a simple LK (E)-module via H(v) the quotient map LK (E) → LK (G). This simple LK (E)-module is denoted by Nv . f f (iii) The Simple Module V[p] : For any infinite path p, V[p] is the twisted simple left L-module obtained from the simple L-module V[p] . See [10] for details. Proposition 2.6. If v is an infinite emitter such that 0 < |s−1 (v) ∩ r−1 (M (v))| < ∞, then BH(v) Sv∞ ∼ . = Nv Proof. Now v ∈ BH(v) . From Proposition 2.5 above and Lemma 3.5 of [10], it is clear that B

both Sv∞ and Nv H(v) are annihilated by the same primitive ideal I(H(v), BH(v) \{v}). B

Also the K-bases of Sv∞ and Nv H(v) are in bijective correspondence. Indeed if p = p e is a path with r(p) = v in E, then, in the graph F considered in defining the simple module B Nv H(v) in paragraph C) (i) above, r(e ) = v  and s(e ) = s(e) = r(p ) and so p e is a path in F with r(p e ) = v  . Then v −→ v  and p = p e −→ p e is the desired bijection. B It is then clear that the map φ : Sv∞ −→ Nv H(v) given by φ(v) = v  and φ(p e) = p e B extends to an isomorphism from the simple module Sv∞ to Nv H(v) . 2 Proposition 2.7. If v is an infinite emitter for which |s−1 (v) ∩ r−1 (M (v))| = 0, then H(v) Sv∞ ∼ = Nv . Proof. This is immediate from Proposition 2.5 and Lemma 3.6 of [10] if one observes that these two simple modules have the same K-basis and the same annihilating primitive ideal. 2 Notation 2.8. Our main focus is the case when v is an infinite emitter for which |s−1 (v) ∩ r−1 (M (v))| is infinite. In this case, we conform to the notation used in [10] and denote the simple module Sv∞ corresponding to such a vertex v by Nv∞ . The next theorem summarizes the main result of this section. Theorem 2.9. Let v be an infinite emitter for which |s−1 (v) ∩ r−1 (M (v))| is infinite. Then module Nv∞ is a new simple left L-module with annihilating primitive ideal I(H(v), BH(v) ) and Nv∞ is not isomorphic to any of the five simple left L-modules of defined above.

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Proof. The simplicity of Nv∞ has been established in Proposition 2.3 and its annihilating primitive ideal is described in Proposition 2.5. For convenience, we denote the five simple modules mentioned earlier as B

f 2) Nw , Nv1H(v1 ) , NH(v , V[p] , V[p] . v2 f f Now Nv∞  V[p] since the annihilator of V[p] is a non-graded primitive ideal ([10], Lemma 3.4) while, as we proved in Proposition 2.5, Ann L (Nv∞ ) = I(H(v),BH(v) ) is a graded ideal. The proof that Nv∞  V[p] uses an argument similar to that of Chen ([14], Theorem 3.7 (3)). We outline the proof for completeness. Suppose ϕ : Nv∞ −→ V[p] is an n L-morphism. We claim ϕ = 0, that is, ϕ(v) = 0. Otherwise, write ϕ(v) = i=1 ki qi where qi ∈ [p] and assume that the qi are all different. Choose n so that τ≤n (qi ) are all pairwise different. Now, in the definition of Nv∞ as an L-module, e∗ · v = 0 for all e ∈ E 1 and so τ≤n (q1 )∗ · v = 0, but ϕ(τ≤n (q1 )∗ · v) = τ≤n (q1 )∗ · ϕ(v) = k1 τ>n (q1 ) = 0, a contradiction. Hence Nv∞  V[p] . B

It was shown in Lemmas 3.1, 3.5 and 3.6 of [10] that the annihilators of Nw , Nv1H(v1 ) H(v ) and Nv2 2 are respectively       I H(w), BH(w) , I H(v1 ), BH(v1 ) \{v1 } and I H(v2 ), BH(v2 ) , all of which are graded ideals. The set of vertices belonging to these annihilating ideals are respectively H(w), H(v1 ) and H(v2 ). Since the annihilator of Nv∞ is also a graded ideal, it is enough if we can show that these sets H(w), H(v1 ), H(v2 ) and H(v) are all different. Now the vertex set H(w) = H(v), since otherwise M (w) = M (v) and this is not possible as M (w) contains a sink (namely, w), while M (v) does not contain any sinks. Hence Nv∞  Nw . Likewise, H(v2 ) = H(v), since otherwise M (v2 ) = M (v) which will imply that v2 ≥ v in M (v2 ) contradicting the fact that v2 is a sink in M (v2 ). B H(v ) So Nv∞  Nv2 2 . Finally, even if H(v1 ) = H(v), the annihilators of Nv1H(v1 ) and Nv∞ are different (being I(H(v1 ), BH(v1 ) \{v1 }) and I(H(v), BH(v) ) respectively) and so B

Nv1H(v1 )  Nv∞ .

2

3. The cardinality of the set of simple LK (E)-modules As before, E shall denote an arbitrary graph with no restrictions on the cardinality of E 0 or E 1 , and the Leavitt path algebra LK (E) will be denoted by L. We wish to estimate the size of the distinct isomorphism classes of simple left LK (E)-modules. In this connection, we follow the ideas of Rosenberg [21]. However, we need to modify his arguments for the case of Leavitt path algebras which, among other differences, do not always have multiplicative identities. We first compute the cardinality of the set of all simple left LK (E)-modules isomorphic to a fixed simple module S and note that it is bounded above by the cardinality of LK (E). Using a Boolean subring of idempotents

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induced by the paths in LK (E), we obtain a lower bound for the cardinality of the set of non-isomorphic simple LK (E)-modules. In particular, if LK (E) is a countable dimensional simple algebra, then it will have either exactly 1 or at least 2ℵ0 distinct isomorphism classes of simple modules. We begin with a simple description of maximal left ideals of L. Lemma 3.1. Suppose M is a maximal left ideal of L. Then for any idempotent  ∈ / M, M  ⊂ M and M can be written as M = N ⊕ L(1 − ) where N = M  = M ∩ L. Every simple left L-module S is isomorphic to Lv/N for some v ∈ E 0 and some maximal L-submodule N of Lv. Proof. Let M be a maximal left ideal of L and let  = 2 ∈ L\M . If x ∈ M ∩ L then x = x and so M ∩ L ⊂ M . By maximality, M ∩ L = M , so M  ⊂ M for all idempotents  in L. Writing each x ∈ M as x = x + (x − x), we obtain M = M  ⊕ M (1 − ) ⊂ M  ⊕ L(1 − ). By maximality, M = N ⊕ L(1 − ) where N = M  = M ∩ L is a maximal L-submodule of L. Suppose S is a simple left L-module, say S = L/M for some maximal left ideal of L.  Since L = v∈E 0 Lv and M = L, there is a vertex v ∈ / M . By the preceding paragraph, we can write M = N ⊕ L(1 − v) where N = M ∩ Lv. Then S = [Lv ⊕ L(1 − v)]/[N ⊕ L(1 − v)] ∼ = Lv/N . 2 Remark 3.2. It is clear from Lemma 3.1 that, to each maximal left ideal M of L, there  is a unique vertex u ∈ / M such that M = (M ∩ Lu) ⊕ v∈E 0 \{u} Lv. Lemma 3.3. Suppose Lv/N is a simple left L-module with v ∈ E 0 . Then, for any vertex u and a maximal L-submodule N  of Lu, the simple module Lu/N  is isomorphic to Lv/N if and only if there is an element a = uav ∈ Lv such that a ∈ / N and N  a ⊂ N . In this  case, N = {y ∈ Lu : ya ∈ N }. Proof. Suppose σ : Lu/N  → Lv/N is an isomorphism. Let σ(u + N  ) = x + N for some x ∈ Lv. Now σ(u + N  ) = σ(u(u + N  )) = u(x + N ) = ux + N . Then a = ux satisfies a = uav, a ∈ / N and σ(u + N  ) = a + N . Moreover, N  a ⊂ N because, for any y ∈ N  , we have ya + N = y(a + N ) = yσ(u + N  ) = σ(y + N  ) = σ(0 + N  ) = 0 + N . Note that the left ideal I = {y ∈ Lu : ya ∈ N } contains N  and I = Lu since u ∈ / I (as ua = a ∈ / N ).   Hence I = N , by the maximality of N . / N and a = uav. Define Conversely, suppose N  a ⊂ N for some a satisfying a ∈ f : Lu/N  → Lv/N by f (y + N  ) = ya + N . Now N  a ⊂ N implies that f is well-defined and is a homomorphism. Also f = 0 since f (u + N  ) = ua + N = a + N = N . As both Lu/N  and Lv/N are simple modules, f is an isomorphism. 2 Lemma 3.4. Let v be a vertex and A = Lv/N be a simple left L-module. Suppose, for u, w ∈ E 0 , B = Lu/N1 and C = Lw/N2 are both isomorphic to A and b = ubv and

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c = wcv are the corresponding elements satisfying b, c ∈ / N , N1 b ⊂ N and N2 c ⊂ N as established in Lemma 3.3. Then B = C implies b = c. Proof. Suppose, on the contrary, b = c. First of all u = w since otherwise b = ub = uc = uwcv = 0, a contradiction. Thus u = w and N1 , N2 are distinct maximal submodules of Lu. Then N1 b ⊂ N and N2 b ⊂ N implies (N1 + N2 )b = Lub ⊂ N . Since b = ub, we get b ∈ N , a contradiction. 2 From the preceding lemmas and Remark 3.2, we get the following theorem describing the size of a single isomorphism class of simple modules containing a specified simple left L-module. Theorem 3.5. Let Lv/N be a fixed simple left L-module, where v ∈ E 0 and N is a fixed maximal L-submodule of Lv. (a) For any given vertex u, the cardinality of the set of all maximal left ideals M of L satisfying u ∈ / M and L/M ∼ = Lv/N is |uLv\N |. ∼ Lv/N (b) The cardinality of the set of all maximal left ideals M of L for which L/M =  is u∈E 0 |uLv\N | (and thus is ≤ |L|). Proof. (a) From Lemmas 3.3 and 3.4, it is clear that for a given vertex u, the cardinality σu of all the maximal submodules N  of Lu such that Lu/N  ∼ = Lv/N is ≤ |uLv\N |. To prove the equality, let a = uav ∈ uLv such that a ∈ / N . Define N  = {y ∈ Lu : ya ∈ N }. Now N  = Lu since u ∈ / N  due the fact that ua = a ∈ / N . Define φ : Lu/N  → Lv/N  by φ(ru + N ) = rua + N . Clearly φ is a well-defined homomorphism and φ = 0, as φ(u) = ua + N = a + N . Also ker(φ) = 0. Because, if rua + N = N , then (ru)a ∈ N , so ru ∈ N  and we get ru + N  = N  . Since Lv/N is simple and φ is monic, φ : Lu/N  → Lv/N is then an isomorphism. In particular, N  is a maximal L-submodule of Lu. In view of Lemma 3.4, we get a bijection between the set uLv\N and the set {N  : N  maximal submodule of Ru with Ru/N  ∼ = Rv/N }. This shows that σu = |uLv\N |. From Remark 3.2, we then conclude that for a fixed vertex u the cardinality of all the set of maximal left ideals M of L for which u ∈ / M and L/M ∼ = Lv/N is σu = |uLv\N |. (b) As noted in Remark 3.2, to each maximal left ideal M of L there is a unique vertex u ∈ / M in which case M = N  ⊕ L(1 − u). We then appeal to part (a) to reach the desired conclusion. 2 For subsequent applications, we obtain the following reformulation of Lemma 3.3 as follows. Lemma 3.6. Let v ∈ E 0 and Lv/N be a fixed simple left L-module. Then, the maximal left ideals M of L for which L/M ∼ = Lv/N are precisely the annihilators in L of non-zero elements a + N of Lv/N with a = uav for some vertex u.

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Proof. Suppose L/M ∼ = Lv/N for some maximal left ideal M of L. By Lemma 3.1, we  can write M = N ⊕L(1−u) where u is a vertex, u ∈ / M and N  = M ∩Lu. By Lemma 3.3, there is an element a = uav ∈ / N so that a +N is non-zero and N  = {y ∈ Lu : ya ∈ N } = {y ∈ Lu : y(a + N ) = N }. It is then clear that M = {r ∈ L : r(a + N ) = N }. Conversely, suppose the left ideal I is the annihilator in L of some non-zero element a + N of Lv/N , where a = uav for some vertex u. Let N  = {y ∈ Lu : y(a + N ) = N } = {y ∈ Lu : ya ∈ N }. From the proof of Theorem 3.5 it is clear that Lu/N  ∼ = Lv/N . Then  M = N ⊕ L(1 − u) is a maximal left ideal of L and M ⊂ I. By maximality, M = I, the annihilator of a + N in L. 2 In the context of Theorem 3.5(b), our next goal is to investigate the size of the set of all non-isomorphic simple left L-modules. Towards this end, we consider maximal left ideals of L that arise from a specified Boolean subring of idempotents in L. A special Boolean subring B of L: Let S = E 0 ∪ {αα∗ : α a finite path in E} ∪ {0}. Observe that elements of S are commuting idempotents. Moreover, if a, b ∈ S, then it is easy to see that ab ∈ S. Let B be the additive subgroup of L generated by S. B is clearly a subring of L. Define, for any two elements a, b ∈ B, a  b = a + b − 2ab and a · b = ab. Then B becomes a Boolean ring under the operations  and · . Define a partial order ≤ on B by setting, for any two elements a, b ∈ B, a ≤ b if ab = a. Then B becomes a lattice under the operations, a ∨ b = a + b − ab and a ∧ b = ab. Proposition 3.7. (a) If M  is a maximal left ideal of L, then M = M  ∩ B is a maximal ideal of B. If M  = N ⊕ L(1 − v) for some vertex v ∈ / M  where N = M  ∩ Lv = M  v, then M = M v ⊕ B(1 − v). (b) Every maximal ideal M of B embeds in a maximal left ideal PM of L such that PM ∩ B = M . Thus different maximal ideals M1 , M2 of B give rise to different maximal left ideals PM1 , PM2 of L. Proof. (a) If M  is a maximal left ideal of L, then clearly M = M  ∩ B is an ideal of B. To show that M is maximal in B, it is enough if we show that M is a prime ideal of the Boolean ring B. Suppose x, y ∈ B such that xy ∈ M and x ∈ / M . Since Lx + M  = L, we can write y = rx + m where r ∈ L and m ∈ M  . Then y = y 2 = rxy + m y. By Lemma 3.1, m y ∈ M  y ⊂ M  and so y ∈ M  ∩ B = M . Thus M is a maximal ideal of B. Let v be a vertex with v ∈ / M  . By Lemma 3.1, M  = N ⊕ L(1 − v) where N = M  v. Note that v ∈ B and M v ⊂ M , as M is an ideal. Thus M = M v ⊕ M (1 − v) ⊂ M v ⊕ B(1 − v). By maximality, M = M v ⊕ B(1 − v). / M . Because (b) Let M be a maximal ideal of B. Then there is at least one vertex v ∈ 0 ∗ ∗ if E ⊂ M , then for every path α with, say s(α) = u, αα = uαα ∈ M , as M is an ideal of B. This implies M = B, a contradiction. We now claim that the left ideal LM v = Lv. k Suppose, by way of contradiction, assume that v ∈ LM v, so that v = i=1 ri mi v where mi ∈ M and ri ∈ L. Observing that m = m1 ∨ · · · ∨ mk belongs to the ideal M and k k satisfies mi m = mi for all i, we get mv = vm = i=1 ri mi mv = i=1 ri mi v = v.

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This is not possible, since mv ∈ M while v ∈ / M . Thus LM v is a proper L-submodule of Lv and hence can be embedded in a maximal L-submodule N of Lv. Writing each element x ∈ M as x = xv + (x − xv) we see that M embeds in the maximal left ideal PM = N ⊕ L(1 − v). By the maximality of M , it is clear that PM ∩ B = M . This implies that if M1 = M2 are maximal ideals of B embedding, as above, in maximal left ideals PM1 and PM2 of L, then PM1 = PM2 . 2 Corollary 3.8. The cardinality of the set of all maximal left ideals of L is at least the cardinality of the set of all maximal ideals of the Boolean ring B. For each maximal ideal M of B, choose one maximal left ideal PM = N ⊕ L(1 − v) where N = PM v as constructed in Proposition 3.7(b). Let T denote the set of all such maximal left ideals PM of L. We shall call each such PM a Boolean maximal left ideal corresponding to the maximal ideal M of B and call the simple module L/PM a Boolean simple module. Note that, by Proposition 3.7(b), |T | = the cardinality of the set of all maximal ideals of B. From Proposition 3.7 it is clear that for each vertex v there is a Boolean maximal left ideal PM = PM v ⊕ L(1 − v) not containing v. Because, given v we can find a maximal left ideal Q of L not containing v. Clearly Q ∩ B = M is a maximal ideal in B not containing v. Then proceed as on Proposition 3.7(b), to construct the Boolean maximal left ideal PM corresponding to M and v. As noted there, PM = PM v ⊕ L(1 − v). The next proposition estimates the size of the set of Boolean simple modules isomorphic to a given simple left L-module. Proposition 3.9. Let Lv/N be a fixed simple left L-module, where v ∈ E 0 and N is a maximal L-submodule of Lv. Let Sv,N = {PM ∈ T : L/PM ∼ = Lv/N }. Then |Sv,N | ≤ dimK (Lv/N ). Proof. By Lemma 3.6, each PMj ∈ Sv,N annihilates an element xj = aj + N ∈ Lv/N . Regarding Lv/N as a K-vector space, we claim that these elements xj (corresponding to the various PMj ∈ Sv,N ) must be K-independent. To justify this, suppose finitely many of these elements xj , with j = 1, ..., n + 1, satisfy n+1 

kj xj = 0

(##)

j=1

where, for each j, kj ∈ K and the maximal ideal PMj is the corresponding annihilator of the element xj . Observe that the maximal ideals of B satisfy the Chinese remainder theorem and so, corresponding to the finite set M1 , · · ·, Mn , Mn+1 of maximal ideals n of B, there is an element b ∈ i=1 Mi such that b ∈ / Mn+1 so that the ideal generated by n n 0 set E ⊂ B, we then see that ( j=1 PMj ) + i=1 Mi and Mn+1 is B. Since the vertex n PMn+1 = L and so there is an element a ∈ j=1 PMj , but a ∈ / PMn+1 . Since a annihilates

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x1 , ···, xn , multiplying Eq. (##) on the left by the element a, we get kn+1 axn+1 = 0 which implies kn+1 = 0. Proceeding like this, we establish the independence of the elements xj . Thus the elements xj can be regarded as part of a basis of Lv/N . Since distinct maximal left ideals PMj correspond to different such elements xj in a basis of LvN (Lemma 3.4), we conclude that |Sv,N | ≤ dimK (Lv/N ). 2 Lemma 3.10. If Soc(L) = 0 and the graph E satisfies Condition (L), then the Boolean ring B is atomless, that is, it has no minimal elements. Proof. Since Soc(L) = 0, the graph E cannot have any line points [12] and, in particular, has no sinks. Suppose, by way of contradiction, B has a minimal element m so that, for all b ∈ B, either mb = 0 or mb = m. In order to reach a contradiction, we first claim that m can be taken to be a monomial of the form γγ ∗ for some path γ. To see this, if m = v is a vertex, then as v is not a sink, it will be the source of some path α and in k that case m = mαα∗ = vαα∗ = αα∗ , a monomial. Likewise, suppose m = i=1 ti αi αi∗ where ti ∈ K and αi αi∗ = αj αj∗ for all i, j. Assume, without loss of generality, that α1 is of maximal length. Then, observing that α1 α1∗ αi αi∗ = 0 implies that α1 α1∗ αi αi∗ = α1 α1∗ , r we conclude that m = α1 α1∗ m = j=1 tij α1 α1∗ = kα1 α1∗ for some k ∈ K. Since m2 = m, we take k = 1 and conclude that m = α1 α1∗ . Thus we can assume that m = γγ ∗ for some path γ with s(γ) = u and r(γ) = w (w may be equal to u). Since w cannot be a line point, there is a vertex in T (w) which is either a bifurcation vertex or is the base of a cycle in T (w). Since every cycle has an exit in E (due to Condition (L)), T (w) will always contain a bifurcation vertex w with w = s(e) = s(f ) for some edges e = f . Denoting a path from w to w by δ, we obtain, by the minimality of m = γγ ∗ ,   γγ ∗ = γγ ∗ (γδe) e∗ δ ∗ γ ∗ . Multiplying on the right by γδf , we get γδf = γδee∗ f = 0, a contradiction. This proves that the Boolean ring B has no minimal elements. 2 Theorem 3.11. Let E be an arbitrary graph satisfying Condition (L). If L = LK (E) has K-dimension ℵ0 and Soc(L) = 0, then L has at least 2ℵ0 distinct isomorphism classes of simple left L-modules. Proof. Consider the Boolean ring B of L defined earlier. By Lemma 3.10, the Boolean ring B has no minimal elements. Thus B is a countable atomless Boolean ring without identity. In this case, it is well-known (see [17] or Theorems 1, 8 and 13 in [23]) that the space X of all maximal ideals of B is a locally compact totally disconnected Hausdorff space with no isolated points. Let X ∗ be the one-point compactification of X obtained by the adjunction of a single non-isolated point to X. Now X ∗ is homeomorphic to the Cantor set (see [23]) and so X ∗ , and hence X, has cardinality 2ℵ0 . This means B has 2ℵ0 distinct maximal ideals. From Corollary 3.8 we conclude that there is a set T of 2ℵ0 distinct Boolean maximal left ideals of L obtained from the maximal ideals of B.

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Now for a fixed maximal ideal P = N ⊕ L(1 − v) ∈ T, let SP = {Q ∈ T : L/Q ∼ = ∼ L/P (= Lv/N )}. By Proposition 3.9, |SP | ≤ dimK (Lv/N ). As L and hence Lv/N has countable K-dimension, |SP | ≤ ℵ0 . Since |T| = 2ℵ0 , L has 2ℵ0 non-isomorphic Boolean simple L-modules of the form L/P where P ∈ T. Consequently, L has at least 2ℵ0 non-isomorphic simple left L-modules. 2 Corollary 3.12. Let E be an arbitrary graph. If L = LK (E) is a simple countably infinite dimensional K-algebra, then L has either exactly 1 or at least 2ℵ0 distinct isomorphism classes of simple left L-modules. Proof. If Soc(L) = 0, then, by simplicity, L = Soc(L) and all the simple left L-modules are isomorphic. Suppose Soc(L) = 0, then the simplicity of L implies that the graph E satisfies Condition (L) (see [1]). We then obtain the desired conclusion from Theorem 3.11. 2 Remark 3.13. The method of proof of Theorem 3.11 breaks down if E is an uncountable graph. Because, unlike the case of a countable atomless Boolean ring, the set of maximal ideals of an uncountable atomless Boolean ring B may not have the desired larger cardinality than |B|, unless some conditions such as completeness of B holds, or if B has an independent subset of cardinality |B| (I am grateful to Professor Stefan Geschke for this remark. See [17] for details). When E is an uncountable graph, the Boolean ring B that we constructed in the proof of Theorem 3.11 need not be complete and also need not have a large enough independent subset. 4. Finitely presented simple modules Let E be a finite graph. It was shown in [10] that every simple left LK (E)-module is finitely presented if and only if distinct cycles in E are disjoint, that is, they have no common vertex. Interestingly, in [5], this same condition on the graph E is shown to be equivalent to the algebra LK (E) having finite Gelfand–Kirillov dimension. Further, Theorem 1 of [6] shows that if the graph E has the stated property, then L = LK (E) is necessarily the union of a finite ascending chain of ideals 0 ⊂ I0 ⊂ I1 ⊂ · · · ⊂ Im = L where I0 is a direct sum of finitely many matrix rings Mn (K) over the field K with n ∈ N ∪ {∞} and for each j ≥ 0, Ij+1 /Ij is a direct sum of finitely many matrix rings Mni (K) over the field K and finitely many matrix rings Mli (K[x, x−1 ]) over the Laurent polynomial ring K[x, x−1 ] with ni , li ∈ N ∪ {∞}. We first strengthen the above necessary condition by showing that, if E has the stated property, then in addition to the above condition on the ideal I0 , we have, for each j ≥ 0, (i) Ij is a graded ideal, (ii) Soc(L/Ij ) = 0, (iii) the ideal Ij+1 /Ij is a finite direct sum of matrix rings over

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K[x, x−1 ] only, say Mli K[x, x−1 ] with li ∈ N ∪ {∞} (and does not involve matrix rings over K) and (iv) Ij+1 /Ij is the largest such ideal in the sense that it contains every other ideal N/Ij of L/Ij which is a direct sum of matrix rings over K[x, x−1 ]. We then show that the converse of this strengthened version of the statement of Theorem 1 of [6] holds. This leads to a complete ideal-theoretic structural characterization of the Leavitt path algebras L of a finite graph E over which every simple left module is finitely presented (see Theorem 4.3 below). Moreover, the extensions Ij+1 /Ij do not split for all j ≥ 1, thus answering a remark mentioned in [6]. In passing, an easy proof of Theorem 2 of [6] is also pointed out. Before we proceed, we recall the definition of the “hedgehog graph” H E induced by a hereditary saturated set H (see [9,16]). Let E be a row-finite graph and let H be a non-empty hereditary saturated set of vertices in E. We then construct the graph H E as follows: / H for all i, r(en ) ∈ H}. Define F (H) = {paths α = e1 · · · en : n ≥ 1, s(ei ) ∈ ¯ α : α ∈ F (H)}. Let F (H) = {¯ Then we define the graph H E = (H E 0 , H E 1 , r , s ) as follows: (1) H E 0 = H ∪ F (H). (2) H E 1 = {e ∈ E 1 : s(e) ∈ H} ∪ F¯ (H). (3) For every e ∈ E 1 with s(e) ∈ H, s (e) = s(e) and r (e) = r(e). ¯ ∈ F¯ (H), s (¯ α) = α and r (¯ α) = r(α). (4) For every α The next lemma from [9] states that every graded ideal of L can be realized as the Leavitt path algebra of a hedgehog graph. Lemma 4.1. (See [9].) Let E be a row-finite graph and I be a graded ideal of LK (E) with (I ∩ E 0 ) = H. Then I ∼ = LK (H E) as non-unital rings. We begin with the following easily derivable lemma which was implicit in [1] and was proved in [15]. Lemma 4.2. Let E be any graph and let H be a hereditary subset of vertices in E. If w ¯ the saturated closure of H, then w ∈ H. is the base of a closed path and if w ∈ H In addition to proving the converse of the strengthened form of Theorem 1 of [6], the next theorem provides, in a consolidated fashion, a complete structural characterization of the Leavitt path algebras L over a finite graph over which every simple left module is finitely presented. Theorem 4.3. Let E be a finite graph and let K be any field. Then the following are equivalent for the Leavitt path algebra L = LK (E):

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(i) No two distinct cycles in E have a common vertex; (ii) Every simple left L-module is finitely presented; (iii) L has finite Gelfand–Kirillov dimension; (iv) L is the union of a finite ascending chain of graded ideals 0 ⊂ I0 ⊂ I1 ⊂ · · · ⊂ Im = L

(∗)

with Hj = Ij ∩ E 0 for j ≥ 0, where I0 = Soc(L) and, for each j ≥ 0, Soc(L/Ij ) = 0 and, identifying, L/Ij with LK (E\Hj ), Ij+1 /Ij is the ideal generated by the vertices in all the cycles without exits in E\Hj ; (v) L is the union of a finite ascending chain of graded ideals 0 ⊂ I0 ⊂ I1 ⊂ · · · ⊂ Im = L

(∗∗)

where I0 is a direct sum of finitely many matrix rings of the form Mni (K) over the field K where ni ∈ N ∪ {∞}, for each j ≥ 0, Soc(L/Ij ) = 0, Ij+1 /Ij is a direct sum of finitely many matrix rings of the form Mli (K[x, x−1 ]) where li ∈ N ∪ {∞} and Ij+1 /Ij contains every other graded deal N/Ij of L/Ij which is a direct sum of matrix rings of the form Mr (K[x, x−1 ]) where r ∈ N ∪ {∞}; (vi) E 0 is the union of a finite ascending chain of hereditary saturated subsets H 0 ⊂ · · · ⊂ Hm = E 0

(∗∗∗)

where H0 is the hereditary saturated closure of all the line points in E and, for each j ≥ 0, E\Hj has no line points and Hj+1 \Hj is the hereditary saturated closure of the set of vertices in all the cycles without exits in the graph E\Hj . Proof. The equivalence of (i) and (ii) was proved in [10] and that of (i) and (iii) was proved in [5]. Our proof of (i) =⇒ (iv) below refines the ideas of the proof of Theorem 1 in [6]. Assume (i). Let I0 = Soc(L), so I0 is the (graded) ideal generated by all the line points in E [12]. Now, for any graded ideal J containing Soc(L), Soc(L/J) = 0. To see this, first note that L/J ∼ = LK (E\H) where H = J ∩ E 0 . Since the hereditary saturated set H contains all the sinks in E and since every vertex in E is regular, the (finite) quotient graph E\H contains no sinks and hence no line points. Consequently Soc(LK (E\H)) = 0. Suppose for n ≥ 0 we have defined the graded ideal In  I0 . Let Hn = In ∩ E 0 . By the preceding statements, Soc(L/In ) = 0 and further the finite graph E\Hn satisfies the same hypothesis as E and has no sinks, so that every vertex in it connects to a cycle. Moreover, we claim that E\Hn contains cycles without exits. To see this, for any two cycles c, c in E\Hn , define c ≥ c if there is a path from a vertex in c to a vertex in c . Since no two cycles in E\Hn have a common vertex, ≥ is antisymmetric and hence is a partial order. Then every cycle which is minimal in this partial order has no exits in E\Hn . Define In+1 /In to be the ideal generated by the vertices in all the

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cycles without exits in E\Hn . Clearly, In+1 is a graded ideal of L. By induction on n, after a finite number of steps, we then obtain the chain (∗) with the desired properties, thus proving (iv). (iv) =⇒ (v). Now the socle I0 is the ideal generated by finitely many line points in the finite graph E and by Theorem 5.6 of [12], I0 is a direct sum of finitely many matrix rings Mn (K) over the field K, where n ∈ N ∪ {∞}. Also, since for each j ≥ 0, Ij+1 /Ij is generated by the vertices in all the cycles without exits in E\Hj and since Soc(L/Ij ) = 0, we appeal to Proposition 3.7 of [4] to conclude that Ij+1 /Ij is a direct sum of finitely many matrix rings of the form Mn (K[x, x−1 ]) where n ∈ N ∪ {∞}. We claim that if a graded ideal N/Ij of L/Ij is a direct sum of matrix rings of the form Mn (K[x, x−1 ]) where n ∈ N ∪ {∞}, then N/Ij  Ij+1 /Ij , thus making Ij+1 /Ij the largest graded ideal of this form. To see this, identifying L/Ij with LK (E\Hj ), first note that, if H is the set of all vertices in N/Ij , then N/Ij ∼ = LK (H (E\Hj )) by Lemma 4.1. So, by Theorem 3.9 of [4], no cycle in H (E\Hj ) has an exit and every vertex in H (E\Hj ) connects to a cycle in H (E\Hj ) (noting that E\Hj and hence H (E\Hj ) has no line points). Also, from the definition of the graph H (E\Hj ), every cycle in H (E\Hj ) is contained in H and has no exit in E\Hj . It is then clear that the graded ideal N/Ij is generated by the vertices in these cycles. Since, by hypothesis, all the vertices in no exit cycles in E\Hj belong to Ij+1 /Ij , we conclude that N/Ij  Ij+1 /Ij . (v) =⇒ (vi). Define Hj = Ij ∩ E 0 for all j ≥ 0 and obtain the chain (∗∗∗). Clearly I0 ⊂ Soc(L). Since Soc(L/I0 ) = 0, I0 = Soc(L) and hence, by [12], H0 is the hereditary saturated closure of all the line points in E. Also Soc(L/Ij ) = 0 for all j ≥ 0 implies that the finite graph E\Hj contains no line points and hence no sinks. So every vertex in the finite graph E\Hj connects to a cycle in E\Hj . Since Ij+1 /Ij is a direct sum of matrix rings of the form Mn (K[x, x−1 ]) where n ∈ N ∪ {∞} and contains all the graded ideals of the same form in L/Ij , applying Lemma 4.1 together with Theorem 3.9 of [4], we conclude that Ij+1 /Ij contains all the cycles without exits in E\Hj and is indeed generated by the vertices in all these cycles. It is then clear that Hj+1 \Hj is the hereditary saturated closure of the set of vertices in all the cycles without exits in the graph E\Hj . (vi) =⇒ (i). Suppose, by way of contradiction, there is a vertex w in E which is the base of two distinct cycles g, h. Now w ∈ / H0 since otherwise, by Lemma 4.2, w will be a line point in E, a contradiction. Let t ≥ 0 be the smallest integer such that w ∈ / Ht , so w ∈ Ht+1 \Ht . Now Ht+1 \Ht is the saturated closure of the set St of all the vertices on cycles without exits in the quotient graph E\Ht . Then, by Lemma 4.2, w ∈ St , a contradiction. This proves (i). 2 Remark 4.4. (a) In [6], the authors remark “it is unclear whether the extensions Ij+1/Ij in chain (∗∗) split”. Observe that our stronger condition states that for all j ≥ 0, Ij+1 /Ij contains every ideal N/Ij of L/Ij which is a direct sum of matrix rings over K[x, x−1 ]. This implies that for all j ≥ 1, Ij is never a direct summand of Ij+1 and thus the extension Ij+1 /Ij never splits.

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(b) However for some graphs E with the given property, I1 /I0 may split and indeed I0 may be a direct summand L. For example, let E be a finite graph where E 0 = {u, v, w}, E 1 = {e, f, g} with s(e) = u, r(e) = v = s(f ), and r(f ) = w = s(g) = r(g). Clearly E satisfies the conditions of Theorem 4.3. Here I0 = u is the socle of L = LK (E) Actually, I0 = Lu ⊕ Lue∗ where Lue∗ ∼ = Lu is simple. Since ee∗ ∈ Lue∗ , Lue∗ = Lee∗ ∗ is a direct summand of L. Since Lee ⊕ Lf f ∗ = Lv, we obtain L = Lu ⊕ Lv ⊕ Lw = Lu ⊕ Lee∗ ⊕ Lf f ∗ ⊕ Lw = I0 ⊕ Lf f ∗ ⊕ Lw showing that I0 is a direct summand of L and hence of I1 . For an example of a graph E for which the extension I1 /I0 does not split, see Proposition 4.6 below. We now illustrate Theorem 4.3 by the simplest example of the algebraic Toeplitz algebra by using the ideas from [10]. Example 4.5. Let E be the graph with two vertices v, w, an edge f with s(f ) = v, r(f ) = w and a loop c with s(c) = v = r(c). Since w is the only line point, the socle of L = LK (E) is S = w (see [12]) and there is an epimorphism L −→ K[x, x−1 ] with kernel S mapping v to 1, c to x and c∗ to x−1 . Thus L/S ∼ = K[x, x−1 ] and, moreover, ∞ S = Lw ⊕ n=0 Lwf ∗ (c∗ )n is the direct sum of infinitely many isomorphic simple left ideals in L. We wish to show that every simple left L-module A = L/M is finitely presented, where M is a maximal left ideal of L. If the socle S  M , then we have a direct decomposition S = (S ∩ M ) ⊕ T and L = M ⊕ T . Writing 1 =  +  with  in M and  ∈ T , we get that M = L is cyclic. Suppose S  M . Since K[x, x−1 ] is a principal ideal domain, there is an irreducible polynomial p(x) = 1 + k1 x + · · · +km xm ∈ K[x, x−1 ] such that the maximal ideal M/S = p(x). So p(c) = v + k1 c + · · · +km cm and M = Lp(c) + S = Lp(c) + ∞ m−1 ∗ ∗ i Lw + n=0 Lwf ∗ (c∗ )n . Let N = i=0 Lf (c ) ⊂ S. Suppose r ≥ m − 1 and that f ∗ (c∗ )t ∈ Lp(c) + Lw + N for all t ≤ r. Then, f ∗ (c∗ )r+1 = f ∗ (c∗ )r+1 p(c) − k1 f ∗ (c∗ )r − · · · −km f ∗ (c∗ )r+1−m ∈ Lp(c) + N . This shows that M = Lp(c) + S = Lp(c) + Lw + N is finitely generated and hence the simple module L/M is finitely presented. Remark. Observe that, in our proof above, we never used the fact that the polynomial p(x) is irreducible. Since K[x, x−1 ] is a principal ideal domain, the same argument then shows that every left ideal A  S in L is a finitely generated left ideal. Also, if A is a left ideal such that S  A and A  S, then decomposing S = (S ∩ A) ⊕ T , we see that the left ideal A + S = A ⊕ T  S. Thus A ⊕ T and hence A is a finitely generated left ideal in L. On the other hand, if A  S, A need not be finitely generated. This is clear if A = S, as S is a direct sum of infinitely many (isomorphic) simple left ideals. Indeed the only non-finitely generated left ideals of L are the left ideals A  S which are direct sums infinitely many simple left ideals of L. As noted in the above example, the socle S of the algebraic Toeplitz algebra L is a direct sum of infinitely many simple modules and L is a ring with identity 1. Then S

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