Rings whose class of projective modules is socle fine

Publ. Mat. 48 (2004), 397–408 RINGS WHOSE CLASS OF PROJECTIVE MODULES IS SOCLE FINE Abdelouahab Idelhadj, El Amin Kaidi, Dolores Mart´ ın Barquero and C´ andido Mart´ ın Gonz´ alez Abstract A class Cof modules over a unitary ring is said to be socle fine if whenever M, N ∈ C with Soc(M)∼ =Soc(N) then M∼ =N. In this work we characterize certain types of rings by requiring a suitable class of its modules to be socle fine. Then we study socle fine classes of quasi-injective, quasi-projective and quasicontinuous modules which we apply to find socle fine classes in special types of noetherian rings. We also initiate the study of those rings whose class of projective modules is socle fine. 1. Introduction The notion of socle fine class of modules has been previously used in the algebraic literature without an explicit formulation. Thus in [5, Theorem 9.3.7, p. 166] it is proved that an algebra Ais quasi-Frobenius if and only if each principal A-module has a simple socle and, for any two nonisomorphic principal A-modules P1and P2, we have Soc(P1)6∼ =Soc(P2). Clearly, this last assertion is equivalent to the fact that the class of principal A-modules is socle fine. Also, the result given in [5, Corollary 9.4.3, p. 171] could be stated by proclaiming that any class of indecomposable modules of the same finite length, over a uniserial algebra, is socle fine. Other results in this vein appear in [15] where some socle fine classes are found in the context of CEP-rings. More recently, Page and Zhou in [23, Theorem 24, p. 2920] find a series of equivalent conditions characterizing the socle fine character of certain natural classes. An explicit formulation of the notion was given by A. Idelhadj and E. A. Kaidi in [11] where they 2000 Mathematics Subject Classification. Primary: 13C13; Secondary: 16D10. Key words. Socle fine class, projective module, QI-module, QP-module. Supported by the Spanish DGICYT with project numbers BFM 2001–2335 and BFM 2001–1886, by the Junta de Andaluc´ıa projects: FQM-336, FQM 0194, and ‘Estudio anal´ıtico-algebraico de Sistemas Triples y de Pares en diferentes clases de estructuras no asociativas’. 398 A. Idelhadj et al. give a socle fine characterization of semiartinian rings. Continuing this philosophy, artinian and noetherian rings are characterized in [16] by socle fine classes. The interesting paper [12] contains also characterizations for V-rings and pseudo-Frobenius rings. Complementary literature on socle and radical fine classes can be found in [13], [17] and [10]. Throughout this work the word ring will mean a unitary (associative) ring and modules are understood as unitary (left) modules. If Ris a ring and Xa class or R-modules, we shall say that Xis socle fine (respectively radical fine) whenever for any M, N ∈Xwe have Soc(M)∼ =Soc(N) (respectively M/ Rad(M)∼ =N/ Rad(N) ) if and only if M∼ =N. We shall denote the injective hull of a module Mby E(M). 2. QI, QP and quasi-continuous modules Our main references for quasi-injective (QI), quasi-projective (QP) and quasi-continuous modules are [22] and [1]. By introducing QI or QP modules into the scene, we obtain another characterization for semisimple rings: Theorem 2.1. For any ring R, the following assertions are equivalent: 1) Ris semisimple. 2) The class of all QP modules is socle fine. 3) The class of all QI modules is socle fine. 4) The class Cof all the finite direct sums of QP modules is socle fine. Proof: If Ris semisimple the class of all R-modules is socle fine hence 1) implies 2), 3) and 4). Suppose 2). As Rand Soc(R) are QP and Soc(R) = Soc(Soc(R)) we have R∼ =Soc(R) implying that Ris semisimple. If 3) holds, since Soc(E(R)) = Soc(R) = Soc(Soc(R)) and both of E(R) and Soc(R) are QI, we have E(R)∼ =Soc(R) is semisimple and so Ris also. Finally suppose 4). We shall prove that Ris semisimple by showing that the class of the QP R-modules is closed under finite direct sums. Let X1,...,Xnbe QP R-modules, and X=⊕iXibe an element in C, let Y:= Soc(X) which is QP by its semisimple character. Since X, Y ∈ C and Soc(X) = Soc(Y), we conclude X∼ =Yhence Xis semisimple and therefore QP. Now the fact that the class of QP modules is closed under finite direct sum implies that Ris semisimple (see [7] and [20]). Let Cbe a socle fine class, Man element in Cand [M] the isomorphism class of M. Then the class C ∪[M] is also socle fine. Moreover if {Mi}i∈I is a collection of elements in C, the class of R-modules C ∪ (∪i∈I[Mi]) is Projective Modules Are Socle-Fine 399 socle fine. It is obvious that if C1and C2are two R-module socle fine classes such that ∀X∈ C1,∀Y∈ C2, Soc(X)6∼ =Soc(Y) then C1∪ C2is a socle fine class. Theorem 2.2. Let Mbe a quasi-continuous R-module and Cthe class of its direct summands. Then Cis socle fine if and only if Soc(M)is essential in M. Proof: Let us suppose that Cis socle fine. As E(M) = E(Soc(M)) ⊕W with Soc(W) = 0 and Mis quasi-continuous, by [22, Theorem 2.8, (4), p. 20] we have that M= (M∩E(Soc(M)))⊕(M∩W), where Soc(M) is essential in M∩E(Soc(M)) and Soc(M)∩W= 0. Since M∩Wis a direct summand of Mwith zero socle and Cis socle fine, M∩W= 0 and Soc(M) is essential in M. Let us suppose now that Soc(M) is essential in M. Let Nbe a direct summand of M. Then Soc(N) is essential in Nhence E(N) = E(Soc(N)). Therefore if N1, N2∈ C, with Soc(N1)∼ =Soc(N2) we have that E(N1)∼ =E(N2) and by [22, Theorem 2.31, p. 34], we have N1∼ =N2. Corollary 2.1. Let Rbe a ring. Then Ris a V-ring if and only if the class Cof its indecomposable QI modules with essential socle is socle fine. The ring Ris a noetherian V-ring if and only if the class Dof its QI modules with essential socle is socle fine. Proof: Let Rbe a V-ring and Mbe an indecomposable QI R-module. Then E(M) is also indecomposable (see [22, Theorem 2.8, p. 20]). Let S be a simple (hence injective) submodule of M. Then E(M) = Sand so M=S. By the previous corollary, the class Cis socle fine. Reciprocally, if Cis socle fine take a simple R-module S. Then E(S) is indecomposable and therefore S, E(S)∈C. Since Soc(S) = Soc(E(S)) we have S∼ =E(S) and so Sis injective. Thus Ris a V-ring. Suppose now that Ris a noetherian V-ring. It is easy to prove that the indecomposable QI modules are semisimple hence they form a socle fine class D. On the other hand, if Dis socle fine, by the proved previous part of this corollary Ris a V-ring. Next we prove that any semisimple module is injective: let M=⊕iSiwith each Sia simple submodule. Then M∈Dand as Soc(E(M)) = Soc(M) = Mwe deduce that E(M) has an essential socle; then M, E(M)∈Dand Soc(M) = Soc(E(M)) hence M∼ =E(M) and so any semisimple module is injective. This implies that the class in [16, Theorem 3] is socle fine and therefore Ris noetherian. 400 A. Idelhadj et al. Corollary 2.2. Let Dbe a noetherian domain with Krull dimension 1 and Cany class of pairwise relatively injective torsion D-modules. Then Cis socle fine. Proof: By Theorem 2.2 it suffices to prove that any element M∈ C has essential socle. Since Mis quasi-injective, each submodule of Mis essential in a summand of M(see [22, Proposition 2.1, p. 18]). Then M=T⊕ Wwhere Soc(M) is essential in T(and Soc(W) = 0). As E(W) is a direct sum of indecomposable injective D-modules, applying the corollary of [26, Theorem 2.32, p. 53] each injective indecomposable module is of the form E(D/P) with Pa prime ideal of D. As Soc(W)=Soc(E(W))=0. Then each indecomposable component E(D/P) of E(W) has zero socle hence Pis not maximal, and so, since the Krull dimesion of Dis one, P= 0. Consequently E(W) = ⊕i∈IQi, with Qi=Q(D) the field of fractions of D. Then T(E(W)) = 0 implying T(W) = 0. Furthermore M=T(M) = T(T)⊕T(W) = T(T)⊂T⊂M, hence M=T,W= 0 and Soc(M) is essential in M. Corollary 2.3. If Dis a Dedekind domain, the class of the indecomposable D-modules of the same (finite) length is socle fine. Proof: This corollary is a consequence of the structure theory of quasiinjective modules over Dedekind domains. We recall that, in this context, any QI module is either injective or a torsion D-module Msuch that for each nonzero prime ideal P, the P-primary component MPof Mis a direct sum of isomorphic modules each one of them being isomorphic to D/P n(n > 0) or to E(D/P) (see [9]). The indecomposable D-modules of finite length are torsion modules (see [21, Theorem 1, p. 49]), by the structure theory for finitely generated modules over a Dedekind domain, they have the form D/P nwith Pa prime nonzero ideal and nagreeing with the length of D/P n. As a consequence, these D-modules are QI. Moreover the D-module (D/P n)⊕(D/Qn) with P6=Q, is also QI hence D/P nand D/Qnare relatively injective [1, Proposition 2.2, p. 15]. Then applying Corollary 2.2, the class of the indecomposable D-modules of the same finite length is socle fine and the corollary is proved. Let Dbe a Dedekind domain and Ta nonzero torsion D-module. Then Thas a direct summand isomorphic either to E(D/P) or to D/P nP for some maximal ideal Pof Dand nP∈N− {0}(see [18]). By using this, it can be proved that Mis a QP (quasi-projective) D-module if and only if Mis either a projective module or a torsion D-module such that for each maximal ideal P, its P-primary component is a direct sum of modules all isomorphic to R/P npfor some np∈N− {0}. It is easy to Projective Modules Are Socle-Fine 401 prove (see [1, Exercise 18, p. 24]) that if Dis a Dedekind domain and Ma finitely generated torsion D-module then Mis QI, if and only if Mis QP, if and only if M∼ =(D/P n1 1)m1⊕(D/P n2 2)m2⊕ · · · ⊕ (D/P nk k)mk, where P1, P2,...,Pkare different maximal ideals of D, and mi, ni∈N− {0} for each i= 1,2,...,k. Proposition 2.1. Let Dbe a Dedekind domain. If Cis a class of QP D-modules with nonzero socle and all of them with isomorphic radical, then Cis socle fine. In particular any class of QI finitely generated torsion modules with isomorphic radicals is socle fine. Proof: We consider Mand Ntwo elements of the class Cwith isomorphic socles. By the previous paragraph M∼ =M i∈I (D/P mi i)(Ai), N∼ =M j∈J (D/Qnj j)(Bj) where Piand Qjare maximal, and Ai,Bjare nonempty sets. By the fact that the socles are isomorphic we have the existence of a bijection σ:I→J such that |Ai|=|Bσ(i)|,Pi=Qσ(i)for all i∈I. On the other hand, after a suitable reordering, we can write N∼ =Li∈I(D/P ni i)(Ai)therefore Rad(M)∼ =M i∈I (D/P mi−1 i)(Ai), Rad(N)∼ =M i∈I (D/P ni−1 i)(Ai) and as their radicals are isomorphic their primary components are isomorphic also. Thus (D/P mi−1 i)(Ai)∼ =(D/P ni−1 i)(Ai). Futhermore their annihilators agree, that is to say, Pmi−1 i=Pni−1 iimpliying mi=nifor all i. 3. Rings whose class of projective modules is socle fine It has been mentioned in the introduction, that the rings whose class of injective modules is socle fine are precisely the semiartinian rings. It is therefore natural to pose the question on the rings with socle fine class of projective modules. One first approach to the problem is given by the next theorem. 402 A. Idelhadj et al. Theorem 3.1. Let Rbe a ring and Fthe class of the free R-modules. Then the following assertions are equivalent: 1) Fis socle fine. 2) Ris an IBN ring with Soc(R)6= 0 and some homogeneous component in Soc(R)has finite length. Proof: Suppose first that Fis socle fine. If Soc(R) = 0 or no component has a finite length one checks immediately that Soc(R)∼ =Soc(RN) which take us to the contradiction R∼ =RN. Next we prove that Ris IBN. If Rn∼ =Rmfor n, m ∈Nthen Soc(Rn)∼ =Soc(Rm). Since Soc(R) = Sk0 i0⊕(⊕i6=i0S(Ii) i) with k0∈N∗, and Si0,Sihomogeneous components, this gives Sk0n i0⊕(⊕i6=i0S(Ji) i)∼ =Sk0m i0⊕(⊕i6=i0S(Hi) i) whence k0n= k0mand so n=m, as required. Let us prove 2) ⇒1). Consider two free modules R(I),R(J)with isomorphic socles and Soc(R) = Sk0 i0⊕ (⊕i6=i0S(Ii) i) with k0∈N∗, and Si0,Sithe homogeneous components. Then: Soc(R(I)) = S(X) i0⊕(⊕i6=i0S(I×Ii) i) Soc(R(J)) = S(Y) i0⊕(⊕i6=i0S(J×Ii) i) where X={1,...,k0} × I,Y={1,...,k0} × J. From the hypothesis that the socles are isomorphic one gets |X|=|Y|hence |I|=|J|. We recall that a ring Ais left pseudo-Frobenius (a left PF ring) if A is an injective cogenerator. Left PF rings are characterized by the next theorem: Theorem 3.2. Let Abe a ring. Then the following statements are equivalent: 1) Ais a left PF ring. 2) The class of projective A-modules is socle fine and Ais a left cogenerator. Proof: If Ais a left PF ring then by definition and [3] and [4] the second assertion holds. Suppose now that Ais a left cogenerator with its class of projective modules being socle fine. Then Soc(A)6= 0 since if Soc(A) = 0 = Soc(0) then A= 0. Thus Soc(A) = ⊕i∈ISi=⊕i∈ISoc(E(Si)). As Ais a cogenerator then each E(Si) embeds in Aand E(Si) is a direct factor of A. Hence E(Si) is projective. We have then that ⊕i∈IE(Si) is a projective A-module. Then since Soc(A) = Soc(⊕i∈IE(Si)) we get A∼ =⊕i∈IE(Si). As Ais of finite type A∼ =⊕n i=1E(Si) for some n∈N, whence Ais injective and as a consequence Ais a left PF ring. Projective Modules Are Socle-Fine 403 Theorem 3.3. Let Abe a ring, then the following properties are equivalent: 1) Ais a QF ring. 2) The class Dof projective or injective modules is socle fine. Proof: If Ais QF, the class Dis just the class of injective modules (and also the class of projective ones by [19, Theorem 13.6.1, p. 352]). Since A is artinian, this class is socle fine by [16, Theorem 2]. Next we prove that if Dis socle fine, then Ais a QF ring. Take Pa projective module, then we have Soc(P) = Soc(E(P)) and P, E(P)∈ D. Consequently P∼ =E(P) and so Pis injective. By [19, Theorem 13.6.1, p. 352], Ais QF. Proposition 3.1. Let Abe a cogenerator ring. Then the following assertions are equivalent: 1) Ais a left QF3 ring. 2) The class of projective A-modules is socle fine. Proof: If Ais a cogenerator left QF3 ring then Ais a PF-ring by [25, p. 55]. Now by Theorem 3.2 the class of projective A-modules is socle fine. The other implication is trivial applying Theorem 3.2. Proposition 3.2. Let Abe a left QF3 ring. Then the following assertions are equivalent: 1) Ais a QF-ring. 2) The class of projective A-modules is socle fine. Proof: If Ais a QF-ring then the class of projective A-modules agrees with the class of injective modules and as Ais an artinian ring, this class is socle fine (see [16, Theorem 2]). Suppose that the class of projective A-modules is socle fine. Let Pbe a projective A-module and E(P) its injective hull. By [8, Corollary II.6, p. 58], both Pand E(P) are projective. Since Soc(P) = Soc(E(P)) this implies P∼ =E(P) whence Pis injective and Ais a QF-ring. 4. Semiperfect rings Let Abe a ring and Jits Jacobson radical. We recall that Ais semiperfect if A/J is semisimple and any idempotent of A/J is of the form e+Jwith ean idempotent of A. It is well known (see for instance [2, Proposition 27.10, p. 306]) that if Ais semiperfect, each complete family of primitive idempotents contains a basic family e1,...,emof A, and all the basic families of Ahave the same cardinality which is called the capacity of A. The A-modules Ae1,...,Aemform a complete irredundant 404 A. Idelhadj et al. family of representatives of projective indecomposable A-modules. The A-modules Ae1/Je1,...,Aem/Jemform a complete irredundant family of representatives of simple A-modules. We shall use the notation Si= Aei/Jeifor all i. If Pis a projective A-module, then Phas an essentially unique decomposition of the type P= (Aei)(I1)⊕ · · · ⊕ (Aem)(Im). A ring Ais called finitely embedded if and only if it has an essential and finitely generated socle. Any artinian ring is finitely embedded but there are rings which are finitely embedded (and even local) but nonartinian (see [24]). Theorem 4.1. Let Abe a finitely embedded semiperfect ring. The following assertions are equivalent: 1) The class of projective A-modules is socle fine. 2) Acontains all its types of simple A-modules, and any projective indecomposable module has a homogeneous socle. Proof: Consider a finitely embedded semiperfect ring Awith capacity m and a basic family e1,...,emof A. Suppose that the class of projective A-modules is socle fine. If m= 1 then assertion 2) follows from the fact that Soc(A)6= 0. Next we take m≥2, and suppose (after a suitable reordering if necessary) that Soc(A)∼ =Sα1 1⊕ · · · ⊕ Sαr rwith αi6= 0 for i∈ {1,...,r}, and r < m. Then there is a t≤rsuch that S1⊕ · · · ⊕ Sr can be embedded in Aei1⊕ · · · ⊕ Aeit. Consequently Soc(Aei1⊕ · · · ⊕ Aeit) = Sβ1 1⊕ · · · ⊕ Sβr rwith βi6= 0 for all i∈ {1,...,r}. The projective A-modules A(N)and (Aei1⊕ · · · ⊕ Aeit)(N)have isomorphic socles hence we have an isomorphism Ae(N) 1⊕ · · · ⊕ Ae(N) m∼ =Ae(N) i1⊕ · · · ⊕ Ae(N) it. Since t < m there is a j∈ {1,...,m}such that j6∈ {i1,...,it}, but on the other hand, the theorem on the uniqueness of the decomposition of projective A-modules implies that Aej∼ =Aeikfor some ik∈ {i1,...,it} which is contradictory. Next we prove that Soc(Aei) is homogeneous for each i. Suppose that Soc(Aek) = Sα1 1⊕ · · · ⊕ Sαq qwith q≥2 and αi6= 0 for all i. Then there exist Aei1,...,Aeinpairwise different and distinct from Aeksuch that n≤m−q, and Sq+1 ⊕ · · · ⊕ Smcan be embedded in Aei1⊕ · · · ⊕ Aein. Consequently Soc(Aek⊕Aei1⊕ · · · ⊕ Aein) = Sβ1 1⊕ · · · ⊕ Sβm mwith each βi6= 0. As (Aek⊕Aei1⊕ · · · ⊕ Aein)(N) is projective and its socle is isomorphic to the socle of the projective A-module A(N)we have an isomorphism Ae(N) k⊕Ae(N) i1⊕ · · · ⊕ Ae(N) in ∼ =Ae(N) 1⊕ · · · ⊕ Ae(N) m Projective Modules Are Socle-Fine 405 which implies that n+ 1 = mby the previously mentioned uniqueness theorem. Since we had n≤m−qand q≥2, then n+ 1 ≤m−q+ 1 ≤ m−1, a contradiction. Suppose now that 2) holds, and denote by Aeσ(i) the unique direct summand of Acontaining to Si. It is clear that i7→ σ(i) is a permutation of {1,...,m}and Soc(Aeσ(i))∼ =Sni iwith ni6= 0. Take Pand Qtwo projective A-modules with P=⊕m j=1Ae(Ij) σ(j)and Q=⊕m j=1Ae(Hj) σ(j). If Soc(P)∼ =Soc(Q) we have that ⊕j(Snj j)(Ij)∼ = ⊕j(Snj j)(Hj)and according to the uniqueness of the homogeneous components of semisimple modules we have (Snj j)(Ij)∼ =(Snj j)(Hj)for all j. The uniqueness of the decomposition of a semisimple module as a direct sum of simple ones implies the coincidence of cardinals: |Ij|=|Hj| whence P∼ =Q. Corollary 4.1. Let Abe ring, of some of the following types: 1) A finitely embedded local ring. 2) A primary ring (that is Ais artinian and A/ Rad(A)is simple). 3) A commutative artinian ring. Then the class of projective A-modules is socle fine. Proof: Suppose that Ais as in the first possibility. Since any local ring is semiperfect and all the simple A-modules are isomorphic, from Soc(A)⊂Awe conclude that Acontains its unique type of simple A-module. As any projective A-module is free and Soc(A) is homogeneous we have that any projective indecomposable module has a homogeneous socle. In the second case, take into account that the simple modules over a simple artinian ring are isomorphic. It is easy to prove that the simple A-modules of the form Aei/Rad(A)eiare also simple as A/ Rad(A)-modules. Thus they are isomorphic as A/ Rad(A)-modules and also as A-modules. In this way there is only one isomorphism class of simple A-modules and the conditions in item 2 of Theorem 4.1 are satisfied. Finally, if Ais commutative and artinian, it splits into a finite direct sum of local artinian rings. The class of projective modules of any of these summands is socle fine (as proved in the first item), and from this it is easy to derive that the class of projective A-modules is socle fine. In the noncommutative case we do not have in general this property. Take for instance a field Kand Athe artinian ring of triangular matrices defined by: A=K0 K K.