Weak Krull-Schmidt for Infinite Direct Sums of Cyclically Presented Modules over Local Rings

Weak Krull-Schmidt for Infinite Direct Sums of Cyclically Presented Modules over Local Rings

AFSHINAMINI(*) - BABAKAMINI(*) - ALBERTOFACCHINI(**)(^a)

1.Introduction.

There is a surprising analogy between the behavior of direct sums of uniserial modules over an arbitrary ring and the behavior of direct sums of cyclically presented modules over local rings. The reason of this analogy lies mainly in the fact that both the endomorphism ring of a uniserial module over an arbitrary ring and the endomorphism ring of a cyclically presented module over a local ring are either local or exactly have two maximal ideals whose residue rings are division rings. This allows us to give acomplete description of the direct-sum decompositions of amodule into uniserial submodules or into cyclically presented submodules (when the base ring is local) exactly using two invariants. These two invariants are the epigeny class and the monogeny class in the case of uniserial modules [4], and are the epigeny class and the lower part in the case of cyclically presented modules over local rings [1]. In both cases, a Weak Krull-Schmidt Theorem for finite direct sums holds. It is now natural to try to see which further parts of the theory of uniserial modules developed in the last decade also holds for cyclically presented modules over local rings.

(*) Indirizzo dell'A.: Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran.

E-mail: [email protected] [email protected]

(**) Indirizzo dell'A.: Dipartimento di Matematica Pura e Applicata, UniversitaÁ di Padova, Via Trieste 63, 35121 Padova, Italy.

E-mail: [email protected]

(^a) Partially supported by Ministero dell'UniversitaÁ e dellaRicerca(Prin 2005

``Perspectives in the theory of rings, Hopf algebras and categories of modules'') and by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica.

2000 Mathematics Subject Classification. Primary 15A33, 16D70. Secondary 16L30.

The first natural question is to see whether the Weak Krull-Schmidt Theorem holds for infinite direct sums as well. For uniserial modules, this question was answered in three papers. First, Dung and Facchini [3] in- dividuated a possible ``pathology'', given by the so-called non-quasismall modules, as a possible source of difficulties. Secondly, Puninski [11] proved the existence of non-quasismall modules. And thirdly, PrÏõÂhoda[9] finally determined the correct form of the Weak Krull-Schmidt Theorem for in- finite direct sums of uniserial modules. In this paper, we answer the cor- responding question for infinite direct sums of cyclically presented mod- ules over a local ring, that is, we prove a Weak Krull-Schmidt Theorem that holds for infinite direct sums of cyclically presented modules over local rings (Theorem 3.1). The case of cyclically presented modules turns out to be much simpler than that of uniserial modules, essentially because cy- clically presented modules are finitely generated, hence small, so that the pathology of non-quasismall modules cannot appear in this setting.

Another natural question is whether a direct summand of a direct sum of uniserial modules (cyclically presented modules over a local ring) is still a direct sum of uniserial (cyclically presented, respectively) modules; see [5, Problems 9 and 10, p. 268]. For infinite direct sums of uniserial modules, the answer is negative, as was proved by Puninski in [10, Theorem 6.3].

The same holds for cyclically presented modules over a local ring, because the example given by Puninski in [11, Proposition 8.1] of a uniserial domain Rwith anon-zero non-invertible elementrand a direct-sum decomposition K(R=rR)^(@0)(R=rR)^(@0), shows not only the existence of auniserial module K that is not quasi-small, but also that there is a countably gen- erated uniserial direct summandKof adirect sum of cyclically presented modules over the local ringR that is not a direct sum of cyclically pre- sented modules. Thus this second question has a negative answer both in the case of uniserial modules and in the case of cyclically presented mod- ules over local rings. Here is another example. In [10, Theorem 6.3], there is an example of a pure projective modulePover auniserial coherent ring that is not direct sum of indecomposable modules. Over a uniserial ring, every finitely presented module is adirect sum of cyclically presented modules. Therefore,Pis another example of a module that is not a direct sum of cyclically presented modules, but is a direct summand of a direct sum of cyclically presented modules.

A third natural question is therefore whether a direct summand of a direct sum offinitely manyuniserial modules (cyclically presented mod- ules over a local ring) is still a direct sum of uniserial (cyclically presented, respectively) modules. In the finite case, PrÏõÂhoda[8] showed that every

direct summand of a direct sum of finitely many uniserial modules is still a direct sum of finitely many uniserial modules. The general case of direct summands of direct sums of finitely many cyclically presented modules over alocal ring is still open.

Direct summands of direct sums of cyclically presented modules are usually called RD-projectivemodules in the literature. Here ``RD'' stands for ``relatively divisible''. As we have seen above, there exist countably generated indecomposable RD-projective modulesKover local rings that are not direct sums of cyclically presented modules and RD-projective modules P over local rings that are not direct sums of indecomposable modules. Nevertheless, every RD-projective module over acommutative local ring is a direct sum of cyclically presented modules [12, Corollary 2].

However, every RD-projective module over an arbitrary ring is a direct sum ofcountably generatedRD-projective modules. In order to see this, notice that if a moduleMis adirect sum ofc-generated modules, wherecis any fixed infinite cardinal, then so is every direct summand of M [5, Theorem 2.47]. Forcˆ @0, we get that every RD-projective module over an arbitrary ring, not necessarily local, is a direct sum of countably gen- erated RD-projective modules. Therefore every RD-projective module over an arbitrary ring is a direct sum of direct summands of direct sums of countably many cyclically presented modules.

As we have already said, in this paper we completely solve the first question posed above, showing that the Weak Krull-Schmidt Theorem holds for infinite direct sums of cyclically presented modules over local rings (Theorem 3.1). In the last section of the paper, we study the monoid of all isomorphism classes of direct sums of cyclically presented modules over alocal ring.

This paper is the natural generalization of our previous article [1], from the finite case to the infinite one. Throughout the paper, we consider unitary right modules over an associative ring with identity. For a ringR, J(R) a nd U(R) will be the Jacobson radical and the set of all invertible elements ofR, respectively.

2.Preliminary results.

Recall that aright module over aring R is said to be cyclically pre- sentedif it is isomorphic toR=aRfor somea2R. The endomorphism ring of anon-zero cyclically presented moduleR=aRis canonically isomorphic toE=aR, whereE:ˆ fr2Rjra2aRgis theidealizerofaRand the right

idealaRofRis atwo-sided ideal in the subringEofR. The following result is proved in [1, Theorem 2.1].

THEOREM2.1. Let a be a non-zero non-invertible element of a local ring R, let E be the idealizer of aR in R, and let E=aR be the endomorphism ring of the cyclically presented right R-module R=aR. Set

I:ˆ fr2Rjra2aJ(R)g and K:ˆJ(R)\E:

Then I and K are completely prime two-sided ideals of E containing aR, the union(I=aR)[(K=aR)is the set of all non-invertible elements of E=aR, and every proper right ideal of E=aR and every proper left ideal of E=aR is contained either in I=aR or in K=aR. Moreover, exactly one of the fol- lowing two conditions holds:

(a)Either the ideals I and K are comparable, so that E=aR is a local ring with maximal ideal(I=aR)[(K=aR), or

(b)The ideals I and K are not comparable, J(E=aR)ˆ(I\K)=aR, and (E=aR)=J(E=aR)is canonically isomorphic to the direct product of the two division rings E=I and E=K.

From now on, for anon-projective cyclically presented module UˆR=aR over alocal ring R, we use the notation I_U;K_U for the two idealsI=aR;K=aRof EndR(U) described in Theorem 2.1.

Recall that two modulesAandB have the same epigeny class(notation [A]_eˆ[B]_e), if there exist an epimorphism A!B and an epimorphism B!A; see [4]. Notice that, if a;b are elements of a local ring R, then [R=aR]_eˆ[R=bR]_eif and only if there existu;v2U(R) withua2bRand vb2aR, or, equivalently, if there exist u;v2U(R) a nd r;s2R with uaˆbrandvbˆas(because any epimorphismR=aR!R=bRis induced by left multiplication by some invertible element ofR). IfR=aRandR=bR are two cyclically presented modules over a local ringR, we say thatR=aR andR=bR have the same lower part, and write [R=aR]_l ˆ[R=bR]_l, if there exist u;v2U(R) a ndr;s2R with auˆrband bvˆsa, equivalently, if there existr;s2Rsuch thatrbRˆaRandsaRˆbR(see [1]). The unique cyclically presented module, up to isomorphism, with the same epigeny class as 0 is 0, and the unique cyclically presented module, up to iso- morphism, with the same epigeny class asR_RisR_R. Similarly for the lower part.

The significance of these concepts for cyclically presented modules over local rings is highlighted by the fact that any cyclically presented module is uniquely determined by its lower part and its epigeny class.

LEMMA 2.2 [1, Proposition 4.2]. Let U and V be cyclically presented modules over a local ring R. Then U V if and only if[U]_lˆ[V]_land [U]_eˆ[V]_e.

For the reader's convenience, we collect in the rest of this section some results that will be used repeatedly in the sequel. The proof of Lemma 2.3 follows from [1, Lemma 5.2], Lemma 2.4 follows from [5, Corollary 4.6], and Lemma2.5 follows from Theorem 2.1.

LEMMA2.3. Let V, V⁰be cyclically presented modules over a local ring R. Suppose that there exists a cyclically presented module U such that [U]_lˆ[V]_l and [U]_eˆ[V⁰]_e. Then there exists a cyclically presented module U⁰ such that VV⁰UU⁰. Moreover, [U⁰]_lˆ[V⁰]_l and [U⁰]_eˆ[V]_e.

LEMMA 2.4 (Cancellation property). Let A;B be right modules over a local ring R and U a cyclically presented right R-module such that AUBU. Then AB.

LEMMA2.5. Let U be a non-projective cyclically presented module over an arbitrary local ring R.

(a)If f and g are two endomorphisms of U such that f 2 I_Un K_Uand g2 K_Un I_U, then f ‡g is an automorphism.

(b) Conversely, suppose that f₁;. . .;f_n are n endomorphisms of U, none of which is an automorphism. If f1‡ ‡fn is an automorphism, then there exist two distinct indices i;kˆ1;2;. . .;n such that f_i2 I_Un K_U and f_k2 K_Un I_U.

LEMMA2.6. Let A and B be non-zero cyclically presented right modules over a local ring R with A non-projective.

(a)If there exist homomorphismsa:A!B andb:B!A withba2 I= A, then[A]_lˆ[B]_l.

(b)If there exist homomorphismsa:A!B andb:B!A withba2 K= _A, then[A]_eˆ[B]_e.

The proof of Lemma 2.6 follows from the fact that, for any a;b2R, every homomorphismR=aR!R=bRis induced by left multiplication by somer2RwithraRbR.

3.Arbitrary families of cyclically presented modules over local rings.

Consider two arbitrary families of cyclically presented modules fU_iji2Ig andfV_jjj2 Jg over alocal ring R. The main result of this section says thatL

i2IU_iL

j2JV_j if and only if the familiesfU_iji2Igand fVj jj2Jghave the same lower parts and the same epigeny classes. This fact was proved for finite families of cyclically presented modules in [1].

THEOREM3.1. LetfU_iji2IgandfV_j jj2Jgbe two families of non- zero cyclically presented modules over a local ring R. ThenL

i2IU_iL

j2JV_jif and only if there exist two bijections s;t:I!J such that[Ui]lˆ[Vs(i)]l

and[U_i]_eˆ[V_t(i)]_efor every i2I.

PROOF. Suppose that there are two bijections s;t:I!J such that [Ui]lˆ[Vs(i)]l and [Ui]eˆ[Vt(i)]e for every i2I. We have to show that L

i2IU_iL

j2JV_j.

The symmetric groupSIconsisting of all bijectionsI!Iacts on the set I in a natural way. Let C be the cyclic subgroup of SI generated by t ¹s2S_I. ThenCacts on the setI. For every elementi2Ilet

[i]ˆ f(t ¹s)^z(i)jz2Zg

denote theC-orbit ofi. Lets([i])ˆ fs(x)jx2[i]g be the image of the orbit [i] viathe bijections.

Fix an element i2I. We claim that L

k2[i]U_k  L

`2s([i])V`. In order to prove the claim, set, for simplicity of notation,i_z:ˆ(t ¹s)^z(i),j_z:ˆs(i_z), U_z:ˆU_iz andV_z:ˆV_jz for everyz2Z. Hence if the orbit [i] is infinite, thens([i]) is infinite, andUnˆUm if and only ifnˆm. If the orbit [i] is finite of cardinalityq, thens([i]) is finite of cardinalityq, a ndUnˆUm

if and only ifnm(modq). Note thatt(i_z)ˆt(t ¹s)^z(i)ˆs(t ¹s)^{z 1}(i)ˆ

ˆs(i_{z 1})ˆj_{z 1}. In this notation, the equality [U_i]_lˆ[V_s(i)]_l for every i2I implies that

[Uz]lˆ[Vz]l

(1)

for allz2Z, and similarly the equality [U_i]_eˆ[V_t(i)]_e implies that [U_z]_eˆ[V_{z 1}]_e

(2)

for allz2Z.

We now prove that there are cyclically presented modules X0;X1;. . .

andY1;Y2;. . .satisfying the following properties for every integern1:

(a)X_{n 1}Y_nV_{n 1}V _n andX_nY_nU_nU _n; (b) [Xn]lˆ[Un]l and [Xn]eˆ[U n]e;

(c) [Yn]_l ˆ[V n]_land [Yn]_eˆ[Vn 1]_e.

The construction of theXi's and theYj's is by induction, as follows. Set X0:ˆU0. Since [U0]_lˆ[V0]_l and [U0]_eˆ[V 1]_e, there is acyclically pre- sented moduleY₁such thatV₀V ₁X₀Y₁, [Y₁]_lˆ[V ₁]_land [Y₁]_eˆ

ˆ[V₀]_e(Lemma2.3). Therefore, [U₁]_eˆ[V₀]_eˆ[Y₁]_eand [U ₁]_lˆ[V ₁]_l ˆ

ˆ[Y₁]_l. Hence, again by Lemma 2.3, there is a cyclically presented module X1such thatU1U 1X1Y1, [X1]lˆ[U1]land [X1]eˆ[U 1]e. ThusX0, X1,Y1have the required properties.

Now suppose n>1 and thatX_t and Y_t satisfying the required prop- erties have already been constructed for every t5n. Since [X_{n 1}]_l ˆ

ˆ[U_{n 1}]_lˆ[V_{n 1}]_land [X_{n 1}]_eˆ[U _n‡1]_eˆ[V _n]_e, by Lemma2.3, there exists acyclically presented moduleYn such thatXn 1YnVn 1V n, [Yn]_lˆ[V n]_l and [Yn]_eˆ[Vn 1]_e. From [U n]_lˆ[V n]_l and [Un]_eˆ

ˆ[Vn 1]_e, it follows tha t [Yn]_lˆ[U n]_l and [Yn]_eˆ[Un]_e. Aga in by Lemma2.3, there exists a cyclically presented module X_n such that X_nY_nU_nU _n, [X_n]_lˆ[U_n]_land [X_n]_eˆ[U _n]_e.

Suppose that the orbit [i] is an infinite set. Then M

k2[i]

UkˆM

n2Z

UnˆU0 M

n1

(UnU n)

!

X0 M

n1

(XnYn)

!

ˆM

n1

Xn 1Yn

… †

M

n1

Vn 1V n

… †

ˆM

n2Z

Vnˆ M

`2s([i])

V`: This proves the claim for the case of an infinite orbit [i].

In order to prove the claim in the case in which the orbit [i] is afinite set withqelements, that is, in order to prove that^qL¹

zˆ0U_z^qL¹

zˆ0V_z, it suffices to apply (1), (2) and the Weak Krull-Schmidt Theorem for finite direct

sums of cyclically presented modules over local rings [1, Theorem 5.3].

Thus the claim is true.

When the indexiruns over all the indices inI, we get that the orbits [i] form apartition of Iinto disjoint countable subsets IˆS

i2I[i] and their images s([i]) form apartition of J into disjoint countable subsets JˆS

i2Is([i]). From the claim we know that L

k2[i]U_k L

`2s([i])V<sub>`</sub> for every

orbit [i], so thatL

i2IU_iL

j2JV_j.

For the proof of the opposite implication in the statement of Theo- rem 3.1, we first prove the following auxiliary lemma.

LEMMA3.2. Let MˆL

j2JAjbe a direct sum of arbitrary modules Ajover a local ring R. Suppose that MˆUB, where U is a non-zero cyclically presented R-module. LetpU:MˆUB!U; pk:MˆL

j2JAj !Akbe the structural projections and e_U:U!MˆUB; e_k:A_k!MˆL

j2JA_j be the embeddings. Then:

(a) If U is non-projective, there exist two indices i;k2J such that p_Ue_ip_ie_U 2 I_Un K_U andp_Ue_kp_ke_U 2 K_Un I_U.

(b) If UR_R, there exists an index k2J such that p_Ue_kp_ke_U is an automorphism of U.

PROOF. AsU_Ris finitely generated, there arej₁;. . .;j_n2Jsuch that UA_j1 A_jn. Therefore, for anyj2Jn fj₁;. . .;j_ng,p_j(U)ˆ0. Set C:ˆ L

j6ˆj1;...;jn

Aj, so that MˆAj1 AjnC. IfeU, pU, eB, pB, ejt,pjt, (tˆ1;. . .;n) a nde_C,p_Care the injections and the projections associated to the two direct-sum decompositionsMˆUBˆA_j1 A_jnC, then p_C(U)ˆ0 and hence

1_U ˆp_Ue_U ˆ p_U e_j1p_j1‡ ‡e_jnp_jn‡e_Cp_C e_U

ˆ pUej1pj1eU‡ ‡pUejnpjneU‡pUe_Cp_CeU

ˆ p_Ue_j1p_j1e_U‡ ‡p_Ue_jnp_jne_U: Then we have two cases:

(a) IfU is non-projective, then by Lemma2.5, there exist two indices i;k2 fj₁;. . .;j_ngsuch thatp_Ue_ip_ie_U 2 I_Un K_U andp_Ue_kp_ke_U 2 K_Un I_U.

(b) IfUR_R, then End_R(U)Ris a local ring, hence there exists an indexk2 fj1;. . .;jngsuch thatpUekpkeU is an automorphism ofU. p

We are now ready to prove the remaining implication in the proof of Theorem 3.1. We may suppose that MˆL

i2IUiˆL

j2JVj with Ui;Vj non- zero cyclically presented modules for everyiandj.

We construct the bijections, the existence of the other bijectiontcan be proved exactly in the same way. For any non-zero cyclically presented moduleU, consider the two subsetsI(U)ˆ fi2Ij[U_i]_lˆ[U]_lgofIand J(U)ˆ fj2 Jj[V_j]_lˆ[U]_lg of J. It is obvious that the I(U)'s and the J(U)'s, when U ranges in all the non-zero cyclically presented modules, form apartition ofIandJ, respectively.

In order to establish the existence of the bijectionsbetween the lower parts of fU_iji2Ig and fV_j jj2Jg, it is sufficient to prove that the cardinalitiesjI(U)jandjJ(U)jare equal for every non-zero cyclically pre- sented moduleU.

Fix anon-zero cyclically presented moduleU. Suppose first that either I(U) or J(U) is a finite set. Without loss of generality we may assume jI(U)j jJ(U)j. Suppose that jI(U)j5jJ(U)j. If I(U)ˆ fi₁;. . .;i_ng, let

fj₁;. . .;j_n‡1gbe asubset ofJ(U) of cardinalityn‡1. Write

MˆU_i1 U_inBˆV_j1 V_jn‡1C;

where

Bˆ M

i6ˆi1;...;in

U_i and Cˆ M

j6ˆj1;...;jn‡1

V_j:

We will show by induction onnthat the submoduleBhas a cyclically pre- sented direct summandVisomorphic toV_jtfor sometˆ1;. . .;n‡1, which gives us a contradiction because by Lemmas 2.6(a) and 3.2 there exists i2In fi1;. . .;ingwith [Ui]_lˆ[V]_lˆ[Vjt]_lˆ[U]_l.

IfURR, thenU_i1;. . .;U_inandV_j1;. . .;V_jn‡1are all isomorphic toRR. By cancellingU_i1;. . .;U_inandV_j1;. . .;V_jnfrom both sides (Lemma2.4), we getBCV_jn‡1.

Now suppose thatU6R_R. For anyk2Iand`2J, letp_k:L

i2IU_i!U_k, p_`:L

j2JV_j !V_`,e_k:U_k !L

i2IU_iande_{`</sub>:V<sub>`}!L

j2JV_j be the canonical projec- tions and injections. Apply Lemma 3.2 to U_i1 and the direct sum MˆL

j2JV_j, so that there isj2 Jsuch thatp_i1e_jp_je_i12 K= _Ui1. By Lemma2.6, we have [U_i1]_eˆ[V_j]_e. One of the following two cases holds.

(1)jis equal to somej_t, sa yj₁. ThusU_i1V_j1(Lemma2.2). Therefore, Ui2 UinBVj2 Vjn‡1C

by Lemma2.4.

(2) j6ˆjt for every tˆ1;. . .;n‡1. Since [Ui1]lˆ[Vj1]l, by Lemma 2.3,Ui1U⁰Vj1Vj for some cyclically presented moduleU⁰. Then, again by Lemma 2.4,

U_i2 U_inBV_j2 V_jn‡1D

for some suitable module D. An easy induction argument shows that after n steps we get the required contradiction.

Now suppose thatI(U) a ndJ(U) are both infinite. By symmetry it is sufficient to prove thatjJ(U)j jI(U)j. For everyk2I(U), define asubset A(k) ofJ(U) by settingA(k):ˆ f`2J(U)jp<sub>`</sub>e_kp_ke_{`</sub> is an automorphism of V<sub>`}g if UR_R, a nd A(k):ˆ f`2J(U)jp<sub>`</sub>e_kp_ke_{`</sub>2 I= <sub>V`}g if U is non-pro- jective. Note thatA(k) is afinite set because there is afinite subsetFofJ withU_kL

j2FV_j, so thatp`e<sub>k</sub>ˆ0 for every`2JnF. We claim that J(U)ˆ [

k2I(U)

A(k):

In order to prove the claim take an index`2J(U). By Lemma3.2 applied to the direct summandV`of MˆL

i2IUi, there exists an index k2Isuch that p`e<sub>k</sub>p<sub>k</sub>e` is an automorphism of V` when UR<sub>R</sub> andp`e_kp_ke`2 I= <sub>V`</sub> whenUis non-projective. Hence in each case we have [U_k]_lˆ[V<sub>`</sub>]<sub>l</sub>ˆ[U]<sub>l</sub> (Lemma2.6), i.e.,k2I(U) a nd`2A(k). This proves the claim.

It follows thatjJ(U)j @0jI(U)j ˆ jI(U)j. HencejJ(U)j ˆ jI(U)jifI(U) andJ(U) are both infinite. This concludes the proof of Theorem 3.1. p

4.Monoids of isomorphism classes.

In [1], we have associated to every local ringRabipartite, non-directed graphGˆG(R), without multiple edges, which describes the behavior of cyclically presented right R-modules. It is constructed in the following way. LetL;EandIbe sets of representatives up to having the same lower part, epigeny class and isomorphism of all non-projective cyclically pre- sented rightR-modules, respectively. The set of vertices ofGis the disjoint union ofLandE, and the set of edges ofGisI. The graphGˆ(L[E;I) is bipartite because there are no edges between two vertices inLor between two vertices inE. An edgeU2Iconnects the verticesV2LandW2Eif and only if [U]_lˆ[V]_l and [U]_eˆ[W]_e. We identifyL;EandIwith the classes of all lower parts, all epigeny classes and all isomorphism classes, respectively, of non-projective cyclically presented right R-modules, so that Lˆ f[U]_ljU is non-projective cyclically presentedg,Eˆ f[U]_ejU

is non-projective cyclically presentedgandIˆ fhUi jUis non-projective cyclically presentedg, a ndhUiis the unique edge between [U]_land [U]_e. The connected components ofGdefine apartition of the setL[Eof vertices of the graphGand a partition of the setIof edges, and the con- nected components ofGare full subgraphs ofG. Afullsubgraph of a graph Gwithout multiple edges is asubgraphG⁰ofGsuch that any two vertices of G⁰adjacent inGare adjacent inG⁰as well. Thus a connected component of Gwill be of the typeCˆ(L_C[E_C;I_C) for suitable subsetsL_C;E_CandI_Cof L;E and I, respectively. We say that two non-projective cyclically pre- sented rightR-modulesUandV are in the same connected componentif there is aconnected componentCofGwithhUi 2I_CandhVi 2I_C.

A graph is called acomplete bipartite graphif there is apartitionX[Y of its set of vertices for whichX6ˆ ;,Y6ˆ ;, there are no edges between any two vertices inX, no edges between any two vertices inY, and exactly one edge between any vertex in X and any vertex in Y. For any local ring R, the connected components Cˆ(L_C[E_C;I_C) of the graph Gˆ(L[E;I) are complete bipartite graphs [1, Proposition 8.1].

LetGˆ(V[W;I) be a bipartite graph and letF:ˆN^(V)₀ N^(W)₀ be the free commutative monoid with free set of generators the disjoint union V[W. The elements ofF are tuples of nonnegative integers, almost all zero, indexed in V[W. We write the elements of F in the form (av)_v2V[(bw)_w2W. Here theav's and thebw's belong toN0and are almost all zero. LetS_G be the submonoid ofF whose elements are all (av)_v2V [ [(bw)_w2W2F with P

v2VC

avˆ P

w2WC

bw for every connected component Cˆ(V_C[W_C;I_C) ofG. For instance, for every edgefv;wgofG, the element f_v;w:ˆ(d_v;v)_v2V[(d_w;w)_w2W2F, wheredis the Kronecker delta, is inS_G. The set of allfv;w, wherefv;wgranges in the set of all edges ofG, is aset of generators for the monoidS_G. Moreover, thef_v;w's are precisely all atoms of the monoidS_G. Recall that a non-zero elementmof acommutative monoidM is anatomofMif for everya;b2M,mˆa‡bimpliesaˆ0 orbˆ0.

Let SCP-R be the class of all right R-modules that are finite direct sums of cyclically presented modules, and SCP-Rthe class of all rightR- modules that are finite direct sums of cyclically presented modules non- isomorphic to R_R. For any module M_R, let hM_Ri:ˆ fN_RjN_R is an R- module isomorphic to M_Rg be the isomorphism class of M_R, and let

V(SCP-R):ˆ fhMRi jMR2SCP-Rg be the monoid of all isomorphism

class of finite direct sums of cyclically presentedR-modules with addition induced by direct sum:hM_Ri ‡ hN_Ri ˆ hM_RN_Ri. Similarly, consider the submonoid V(SCP-R):ˆ fhM_Ri jM_R2SCP-Rg of V(SCP-R). The fact

that everyR-module in SCP-Ris uniquely of the formPNwithPfree and N2SCP-R implies that V(SCP-R)N0V(SCP-R), where N0

corresponds to the finitely generated free modules, that is, to the cyclic submonoid ofV(SCP-R) generated byhR_Ri.

By [1, Theorem 5.3], there is asubdirect embedding V(SCP-R)!

!N^(E)₀ N^(L)₀ . HereEandLare the sets of all epigeny classes and all lower parts of non-projective cyclically presented rightR-modules. For the graph Gˆ(L[E;I), the monoids V(SCP-R) a nd S_G are isomorphic [1, Theo- rem 8.3], so thatV(SCP-R)N₀S_G

As we have recalled above, for a local ringR, the graphGˆ(L[E;I) is the disjoint union of its connected componentsC, which are complete bi- partite graphs [1, Proposition 8.1]. It follows that SGˆL

C SC. Now a complete bipartite graph C is completely determined by two cardinal numbers ab1. More precisely, for every pair of cardinal numbers ab1, letC(a;b) be the complete bipartite graph with set of vertices the disjoint union ofaandband exactly one edge between eacha2aand eachb2b. Then, for every complete bipartite graph C, there is one pair (a;b) of cardinal numbers ab1 with CC(a;b). Notice that the commutative monoid S_C(a;b) is a free commutative monoid if and only if bˆ1, and in this caseS_C(a;b)is isomorphic toN^(a)₀ . More precisely, we have the following.

LEMMA 4.1. For cardinal numbersab1,gd1, the monoids S_C(a;b)and S_C(g;d)are isomorphic if and only ifaˆgandbˆd.

PROOF. A monoid isomorphismS_C(a;b)!S_C(g;d) sends the setXof all atoms ofS_C(a;b)onto the setYof all atoms ofS_C(g;d). The atoms ofS_C(a;b)are the elements of the form f_a;b, where a ranges in a and b ranges in b.

Equivalently,Xcorresponds to the set of all edges of the graphC(a;b). Let Fbe the family of all subsetsX⁰ofXsuch that the divisor-closed submonoid hX⁰iofS_C(a;b) generated byX⁰is afree monoid with free set of generators X\ hX⁰i. Recall that the divisor-closed submonoid ofS_C(a;b)generated byX⁰ is the set of all elements s2S_C(a;b) such that there exist n1, x⁰₁;. . .;x⁰_n2X⁰ andt2S_C(a;b) withs‡tˆx⁰₁‡. . .‡x⁰_n. Notice that iff_a;b andfc;dare two atoms in a subsetX⁰2 F, then the two corresponding edges fa;bg and fc;dg are incident, because otherwise there is a relation f_a;b‡f_c;dˆf_a;d‡f_c;bbetween elements ofX\ hX⁰i. It follows that the ele- ments X⁰2 F correspond to subgraphs of C(a;b) that are stars. If we partially orderF by inclusion, then the maximal elements ofF have car- dinality eitheraorb. Similarly forSC(g;d). Hence the isomorphism type of

SC(a;b) determines a and b, that is, SC(a;b)SC(g;d) implies fa;bg ˆ fg;dg.

Fromab1 a ndgd1, it now follows thataˆgandbˆd. p Therefore S_GN^(@)₀ L

i2IS_C(ai;bi)

for acardinal number @ and an indexed set f(a_i;b_i)ji2Ig of pairs (a_i;b_i) of cardinal numbers with a_ib_i2. The following proposition shows that the cardinal number @ and the indexed set of pairs (a_i;b_i);i2I;in this direct-sum decomposition of the commutative monoidS_Gare uniquely determined by the ringR.

PROPOSITION 4.2. If there is a monoid isomorphism C:N^(@)₀ L

i2IS_C(ai;bi)

N^(@₀⁰⁾ L

j2JS_C(a⁰

j;b⁰_j)

for cardinal numbers @;@⁰ and aib_i2 (i2I) and a⁰_jb⁰_j2 (j2 J), then @ ˆ @⁰ and there is a bijectionf:I!J such that(ai;b_i)ˆ(a⁰_f(i);b⁰_f(i))for everyi2I.

PROOF. The monoid isomorphismCsends the set of atoms onto the set of atoms. Since there are no nontrivial relations between the atoms con- tained inN^(@)₀ and every atom in L

i2IS_C(ai;bi)

is subject to some nontrivial relation because a_ib_i2, it follows that @ ˆ @⁰ and that the monoid isomorphism C restricts to an isomorphism L

i2IS_C(ai;bi) L

j2JS_C(a0 j;b⁰_j). We show that for anyi2I, there is auniquej2JwithC(S_C(ai;bi))S_C(a⁰

j;b⁰_j). By considering the same property forC ¹, we conclude thatCrestricts to an isomorphismSC(ai;b_i)S_C(a⁰

j;b⁰_j). Thus by Lemma4.1,aiˆa⁰_jandb_iˆb⁰_j. For afixedi2I, leta;a⁰2a_iandb;b⁰2b_ibe distinct elements. There exists j2 J such that C(f_a;b)2S_C(a0

j;b⁰_j). Since f_a;b‡f_a⁰_;b⁰ ˆf_a;b⁰‡f_a⁰_;b, we infer that C(f_a;b);C(f_a;b⁰);C(f_a⁰_;b) a nd C(f_a⁰_;b⁰) must be in the same com- ponent, i.e., in S_C(a⁰

j;b⁰_j). Now for any a⁰⁰2a_i andb⁰⁰2b_i, there is (x;y)2 2 f(a;b);(a;b⁰);(a⁰;b);(a⁰;b⁰)gsuch thatx6ˆa⁰⁰andy6ˆb⁰⁰. Again from the relation fx;y‡fa⁰⁰;b⁰⁰ ˆfx;b⁰⁰‡fa⁰⁰;y we conclude that C(fa⁰⁰;b⁰⁰)2S_C(a⁰

j;b⁰_j) be- causeC(fx;y)2S_C(a⁰

j;b⁰_j). Therefore,C(S_C(ai;bi))S_C(a⁰

j;b⁰_j), as desired. p REMARK4.3. (a)The previous results in this Section hold not only for the graph that describes the cyclically presented modules over a local ring, which we have introduced in this paper, but also for the graph that de- scribes the uniserial modules over an arbitrary ring, which PrÏõÂhodaand the third author introduced in [6, 7]. The following fact, on the contrary, ap- plies only to our graph and not to that introduced in [6, 7]. LetRbe alocal ring andGˆ(L[E;I) be our graph. HereIis the set of all isomorphism classes of non-projective cyclically presented rightR-modules. LetJ(R)

be the setJ(R)n f0g. Recall that two cyclically presented rightR-modules R=aR, R=bR (a;b2J(R)) are isomorphic if and only if there exist u;v2U(R) withaˆubv[1, Lemma2.4]. Thus there is aright action of the groupU(R)^opU(R) on the setJ(R), andIturns out to be the set of all orbits (a):ˆU(R)aU(R) with respect to this group action. With this point of view,E, the set of all epigeny classes of non-projective cyclically pre- sented rightR-modules, is the quotient set ofIˆ f(a)ja2J(R)gwith respect to the equivalence relation on I defined by (a)(b) if [R=aR]_eˆ[R=bR]_e, that is, if and only if there exist u;v2U(R) a nd r;s2Rwithuaˆbrandvbˆas. Similarly,Lturns out to be the quotient set ofIwith respect to the equivalence relationonIdefined by (a)(b) if [R=aR]lˆ[R=bR]l, that is, if and only if there exist u;v2U(R) a nd r;s2Rwithauˆrbandbvˆsa.

(b) The graphG(R) that describes cyclically presented modules over a local ringRis the same for right modules and left modules, that is, it is a left/right symmetric invariant ofR.

Clearly, the graphGˆG(R) and the monoidSGdescribe the behavior of finite direct-sums of cyclically presented modules over a local ringR, but in view of our Theorem 3.1 the constructions of the graph G and the monoidS_Gcan be extended to arbitrary cardinalities in the following way.

Fix alocal ringRand an infinite cardinal number@. Let SCP_@-Rbe the class of all rightR-modules that are direct sums of families of cardinality5@

of cyclically presented R-modules, and SCP@-R the class of all right R- modules that are direct sums of families of cardinality5@of cyclically pre- sented modules non-isomorphic toR_R. Thus SCP-RˆSCP_@0-Rand SCP- RˆSCP_@0-R. Let V_@:ˆV(SCP_@-R) a nd V_@:ˆV(SCP_@-R) be the corresponding monoids of all isomorphism classes of modules in SCP_@-R and SCP@-R, respectively. ThusV_@ˆL

i2IR=aiR

:jIj5@; ai2R and V_@ˆL

i2IR=a_iR

:jIj5@; a_i2Rn f0g .

LEMMA4.4. Every R-module inSCP_@-R is uniquely of the form PN with P free and N2SCP_@-R, that is, if PNP⁰N⁰with P;P⁰free and N;N⁰inSCP@-R, then PP⁰and NN⁰.

PROOF. Suppose thatPˆL

i2IU_iandP⁰ˆL

j2JU_j⁰, whereU_iU_j⁰R_R for every i2Iand j2J. Also suppose thatNˆ L

k2KV_k andN⁰ˆL

l2LV_l⁰, where theV_k's and theV_l⁰'s are non-projective cyclically presented modules andPNP⁰N⁰. An application of Theorem 3.1 to the isomorphism

PNP⁰N⁰ gives that there are two bijections s;t:I[K!J[L corresponding to the lower parts and epigeny classes. Since a non-projec- tive cyclically presented module does not have the same lower part or epigeny class asR_R,sandtinduce bijectionsI!JandK !L. Now again

by Theorem 3.1, we havePP⁰andNN⁰. p

From Lemma4.4, it follows that V_@[0;@[V_@, where [0;@[ is the commutative additive monoid of all cardinal numbers5@.

For an arbitrary bipartite graphGˆ(V[W;I) and an infinite cardinal

@, letP:ˆ[0;@[^V[0;@[^Wbe the direct product of afamily of copies of the additive monoid [0;@[ whose cardinality is equal to the cardinality of the disjoint union V[W. The elements ofP are tuples of cardinals 5@ in- dexed in V[W. Write the elements ofP in the form (a_v)_v2V[(b_w)_w2W, where theav's and theb_w's are cardinals5@. LetS_G;@be the submonoid of Pwhose elements are all (av)_v2V[(b_w)_w2W2Psubject to two conditions:

(1) P

v2VC

a_vˆ P

w2WC

b_w for every connected componentCˆ(V_C[W_C;I_C) of G; (2) P

v2Vav5@. Notice that the sum of any set of cardinals is defined as the cardinality of their disjoint union.

THEOREM 4.5. For the graph Gˆ(L[E;I) of a local ring R, the monoids V_@and S_G;@are isomorphic. Thus V_@[0;@[S_G;@.

PROOF. By Theorem 3.1, there is awell-defined subdirect embedding F:V_@!Pˆ[0;@[^L[0;@[^Edefined as follows. Let MˆL

j2JU_j 2SCP_@- R. For any non-projective cyclically presented module U let L_U(M)ˆ

ˆ fj2J:[Uj]lˆ[U]lg and EU(M)ˆ fj2J:[Uj]eˆ[U]eg. Then the EU(M)'s and theLU(M)'s form two partitions ofJ. Define

F(hMi)ˆ(jL_U(M)j)_[U]l2L[(jE_U(M)j)_[U]e2E: Hence it suffices to show that the image ofFisS_G;@. Now ifAˆL

j2JR=a_jRis adirect sum of non-projective cyclically presented modulesR=a_jR(jJj5@), the partition of G into its connected components C induces apartition fJ_CjCis aconnected component ofGgofJ, so thatAˆL

C

L

j2JC

R=a_jR . Thus ifF(hAi)ˆ(a_l)_l2L[(b_e)_e2E, then, for every connected componentCof G, P

l2LC

a_l ˆ jJ_Cj and P

e2EC

b_eˆ jJ_Cj. It follows thatF(hAi) is inS_G;@. Con- versely, let (a_l)_l2L[(b_e)_e2E be an element of S_G;@. For every connected componentCofG, P

l2LC

a_lˆ P

e2EC

b_eby Condition (1), so that there exists a

setJCof cardinalityjJCj ˆ P

l2LC

alˆ P

e2EC

b_ewith two mappingslC:JC!LC

andh_C:J_C !E_Csuch that for everyl2L_Cthe inverse image oflinJ_Cvia lChas cardinalityal, and for everye2ECthe inverse image ofeinJCviah_C has cardinalityb_e(this follows immediately from the definition of sum of a set of cardinals recalled above). Since the connected component C is a complete bipartite graph, for everyj2J_C there is anon-projective cycli- cally presented moduleR=a_jRwith epigeny classh_C(j) and with lower part lC(j). Then A:ˆL

C

L

j2JC

R=ajR

is in SCP@-R by Condition (2), and

F(hAi)ˆ(a_l)_l2L[(b_e)_e2E. p

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Manoscritto pervenuto in redazione il 3 giugno 2008.