Rings whose modules have maximal submodules

Publicacions Matem`atiques, Vol 39 (1995), 201–214. RINGS WHOSE MODULES HAVE MAXIMAL SUBMODULES Carl Faith Dedicated to Laci Fuchs on his 70th birthday Abstract A ring Ris a right max ring if every right module M= 0 has at least one maximal submodule. It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module Eof modR; also it suffices to check the submodules of the injective hull E(V) of each simple module V(Theorem 1). Another test is transfinite nilpotence of the radical of Ein the sense that radαE= 0; equivalently, there is an ordinal αsuch that radα(E(V))=0 for each simple module V. This holds iff each radβ(E(V)) has a maximal submodule, or is zero (Theorem 2). If follows that Ris right max iff every nonzero (subdirectly irreducible) quasi-injective right R-module has a maximal submodule (Theorem 3.3). We characterize a right max ring Rvia the endomorphism ring Λ of any injective cogenerator Eof mod-R; namely, Λ/L has a minimal submodule for any left ideal L= annΛMfor a submodule (or subset) M=0ofE(Theorem 8.8). Then Λ/L0has socle = 0 for: (1) any finitely generated left ideal L0= Λ; (2) each annihilator left ideal L0= Λ; and (3) each proper left ideal L0=L+L, where L= annΛMas above (e.g. as in (2)) and Lfinitely generated (Corollary 8.9A). HAMSHER MODULES A module Mis a Hamsher module provided each submodule S=0 has a maximal submodule.1 1Hamsher modules are called max modules by Shock [S]. 202 C. Faith 1. One-Module Theorem. A ring Ris a right max ring iff R has a cogenerating right Hamsher module E. A n.a.s.c. for this is that the injective hull E(V)of each simple right R-module Vis a Hamsher module. Proof: A module Ecogenerates the category mod-Rof all right Rmodules iff for every module M= 0, there is a nonzero map h:M→E ([F1, pp. 91, 148 & 165]). Then h(M)=Mis a nonzero submodule of E. Thus, when Eis a Hamsher module, then Mhas a maximal submodule M, so h−1(M) is a maximal submodule of M. This proves the first statement in Theorem 1. Next let E=⊕E(V), as Vrange over all simple R-modules. Then Eis a cogenerator module for mod-R([F1, p. 167, prop. 3.55]). Let PVbe the projection E→E(V). Then, in the above, 0 =M=h(M)⊆Eimplies 0 =PVh(M)=MV⊆ E(V) is a nonzero submodule of E(V) for some V, and so Mhas a maximal submodule, as before, whenever E(V) is a Hamsher module for all V. Note. E(V) is direct summand of any cogenerator Eof mod-R, hence the Hamsher condition on E(V) is a consequence of that on Ein Theorem 1. Moreover, this is sufficient for Eto be Hamsher. 1.1. Corollary. If Ris a ring such that each simple module Vhas Noetherian injective hull E(V), then Ris a right max ring. To illustrate when E(V) is not only Noetherian, but simple we will cite a theorem of Kaplansky, but first we recall some terminology: Ris right V-ring in case Rhas the equivalent properties. (See [F1, p. 356, 7.32A].) (V1) Every simple right R-module Vis injective, that is, E(V) is simple. (V2) rad M= 0 for each right R-module M. (V3) Every right ideal I=Ris the intersection of maximal right ideals, that is, rad(R/I)R=0. Note. A right V-ring is a right max ring since rad M=Mfor every M=0. Kaplansky’s Theorem. 2A commutative ring Ris a V-ring iff R is Von Neuman regular (= VNR). 2According to my inquiry of Professor Kaplansky, “It worked its way into the public domain” (Letter of October 12, 1994). Rings whose modules have maximal submodules 203 Let J= rad R. Then Jis left vanishing (= T-nilpotent in [B], [H]) if for every sequence {an}∞ n=1 of elements of A, there is an n≥1 so that an···a1= 0, that is the left-hand partial product an···a1vanishes. First Max Theorem ([H], [K]).A commutative ring Ris a max ring iff R/J is VNRand J= rad Ris vanishing. Expressed otherwise: Ris a max ring iff R/J is a V-ring, and J is vanishing. The radical series radα(M) is defined inductively for each ordinal αin the usual way, where rad(M) is the intersection of all maximal submodules of M, radα+1(M) = rad(radα(M)) for any ordinal, and radβ(M)=  α∈β radα(M) for each limit ordinal β. Second Max Theorem ([H], [K]).A ring Ris right max iff R/J is right max and Jis left vanishing. We next show that the modules in the radical series are test submodules for a Hamsher module. 2. Theorem. 3The f.a.e.c.’s on a right R-module M. (1) Mis Hamsher. (2) radβ(M)has a maximal submodule, or is 0, for every ordinal β. (3) radα(M)=0for some α. Proof: (1) ⇒(2) is obvious, and (2) ⇒(3) follows by cardinal number theory for any αof cardinal greater than that of R. (3) ⇒(1). If S=0 is a submodule of M, then Sradλ(M) for least ordinal λ<α, and obviously λis not a limit ordinal, so S⊆radλ−1(M). If S= radλ−1M, then Shas a maximal submodule since rad S= radλ(M)=S. And if S= radλ−1(M), then Sis not contained in a maximal submodule M of radλ−1(M), hence S∩Mis a maximal submodule of S. This proves that Mis a Hamsher module. 3.1. Corollary. Let Ebe a right cogenerator module for R.The Ris right max iff Ehas transfinite nilpotent radical. A n.a.s.c. for 3The equivalence (1) ⇔(3) is a theorem of Shock [S] who also proved that every semi-Artinian Hamsher module is Noetherian. 204 C. Faith this is that E(V)have transfinite nilpotent radical for each simple right R-module V. 3.2. Lemma. If Mis a quasi-injective right R-module, then so is every fully invariant submodule, in particular, so is radα(M), for each ordinal α. Proof: A theorem of Wong-Johnson ([W-J]) characterizes a quasiinjective module as the fully invariant submodules of their injective hulls (see, e.g. [F2, p. 63, Prop. 19.2]). For example, if E=E(M) has endomorphism Λ, then Mis quasi-injective iff λ(M)⊆M∀λ∈Λ. Now let M0be a fully invariant submodule of M. Since E0=E(M)⊆E, and since Eis injective, then every element λ0∈Λ0= End E0is induced by an element λ∈Λ. Since λinduces an endomorphism ¯ λin M, and since ¯ λ(M0)⊆M0by the hypothesis that M0is fully invariant in M, then λ0(M0)⊆M0for each λ0∈Λ0, that is, M0is fully invariant in E(M0), hence is quasi-injective. It follows that radα+1(M) is quasi-injective for all α, since radα+1(M) is fully invariant in radα(M) which by an inductive hypothesis may be assumed to be quasi-injective. Furthermore, radβ(M) is fully invariant hence quasi-injective for each limit ordinal β, since it is the intersection of fully invariant submodules of M. 3.3. Theorem. For a ring R, the f.a.e.c.’s: (1) Ris right max. (2) Every nonzero quasi-injective module has a maximal submodule. (3) Every nonzero subdirectly irreducible quasi-injective module has a maximal submodule. Proof: (1) ⇒(2) ⇒(3) is trivial, and (3) ⇒(1) is an immediate consequence of Theorem 2, Corollary 3.1 and Lemma 3.2. 4. Corollary. If a right Rmodule Mis faithful and has transfinite nilpotent radical, then Rhas transfinite nilpotent radical J. Proof: One shows inductively that radα(M)⊇MJα, where J= rad R. Note. Let Rbe a commutative Noetherian ring. Then Jω=0by the Krull intersection Theorem and if Ris a domain, then Iω= 0 for any ideal I=R([Z-S, p. 216, Theorem 12 and Corollary]). Thus, Jis transfinite but not T-nilpotent when Ris e.g., a Noetherian local domain not a field. Rings whose modules have maximal submodules 205 LOEWY SERIES AND TRANSFINITE SEMISIMPLE MODULES Adescending or dual Loewy series for a module Mis descending chain {Mα}α∈Λof submodules indexed by an ordinal Λ such that M0= M, and Mα/Mα+1 is semisimple Mβ=∩α∈βMα for any limit ordinal β∈Λ. We say that Mis transfinitely semisimple if there is a descending Loewy series {Mα}with Mα= 0 for some α∈Λ. 5. Theorem. Any transfinitely semisimple module Mis a Hamsher module. Proof: By transfinite induction, Mα⊇radα(M) for each Mαas defined above, hence radα(M) = 0 for some ordinal α, and Theorem 1 applies: Mis Hamsher module. By Theorem 1, we also have the following: 5.1. Corollary. If E(V)is transfinite semisimple for each simple right R-module V, then Ris right max. BASS MODULES Recall that a module Mis a Bass module ([F2]) if every submodule M=Mis contained in a maximal submodule of M. 6. Theorem. Let Ebe an quasi-injective right R-module that contains a copy of each simple image of Eand Λ = End ER.IfEis a Bass module, then Λhas essential left socle, socΛ. Proof: By the Harada-Ishii ([H-I]) double annihilator condition (= DAC) for a quasi-injective modules, annΛannEI=I for finitely generated left ideals of Λ,one can show that each such I=0 contains a minimal left ideal L. For if Eis a maximal submodule, containing annEIthe fact V=E/E→Eyields λ∈Λ such that λE ≈V, hence L=Λλis a minimal left ideal contained in I. Thus, socΛ is an essential left ideal of Λ. In the next corollary, we see what happens to Λ when Eis Noetherian. 206 C. Faith 6.1. Corollary. If Eis a Noetherian quasi-injective right module over R, then Λ = End ERis a right perfect ring, hence a right max ring. Proof: By the Harada-Ishii DAC cited in the proof of Theorem 6, ER Noetherian implies that Λ satisfies the DAC on finitely generated left ideals, hence Λ is right perfect ([B]). DOUBLE ANNIHILATOR CONDITIONS FOR COGENERATORS It is known that any cogenerator Fsatisfies the double annihilator conditions (DAC) I= annRannFI (see, e.g. [F1]). We next prove another DAC for F. 7. Dac Theorem. 4If Fis any right cogenerator of R, and Iand M are submodules of RRand FRrespectively, then they satisfy the DAC’s: I= annRannFI(a) M= annFannΩM(b) where Ω = End FR. Proof: (1) Since Fis a cogenerator then R/I →Fαfor some cardinal α, and if (xi) is the image in Fof the coset 1 + Iin R/I, one sees that I= annR{xi}, so (a) follows. (2) F/M embeds in a direct product Fαof copies of F, and hence there is a map h:F→Fαthat has ker h=M. Then, if pα: Fα→Fis the α-th projection, it follows that ωα=pα◦h∈Ω and that (3) M=∩αker ωα. Then, (4) M= annFL, where L=Σ αΩωα. Since (4) =⇒(b), the proof is complete. 4After this was written, I found Kurata’s report [Ku] where (b) is stated without proof in greater generality. Rings whose modules have maximal submodules 207 INJECTIVE COGENERATORS If any cogenerator of mod-Ris a Hamsher module, then Ris a right max ring. In this section we list two conditions on a minimal injective cogenerator Ethat are each necessary and sufficient in order that Rbe a right V-ring: (1) rad E= 0. (Theorem 8.1) and (2) ERis a Bass module, and Λ = End ERhas zero Jacobson radical (Theorem 8.2). 8.1. Theorem. Let Ebe a minimal injective cogenerator of R, and Wthe direct sum of a complete set of non-isomorphic simple right Rmodules. (Thus, Eis the injective hull of W, and Wis the socle of E.) Then, the f.a.e.c.’s: (1) Ris a right V-ring. (2) rad E=0. Proof: (1) ⇒(2). As stated, (1) ⇔rad M= 0 for every right Rmodule M. (2) ⇒(1). If Vis a simple submodule of E, then (2) implies that there exists a maximal submodule Mof Enot containing V. Then since V∩M= 0, and V+M⊃M, we see that E=V⊕M, so Vis injective. Since every simple right R-module embeds in E, then Ris a right V-ring. 8.2. Theorem. If the right minimal injective cogenerator Eof a ring Ris a Bass Module, and if Λ = End ERhas zero Jacobson radical, then Ris a right V-ring (and Eis semisimple). Proof: Let W=socE, the sum of all simple module, one for each isomorphy class. If W=E, then every submodule of Eis a direct summand, hence is injective, so Ris right V-ring. We may therefore assume that E=W, and hence by our Bass module assumption that there is a maximal submodule Mof Ethat contains W. Since V= E/M →W, there is an endomorphism λof Esuch that ker λ=M. Since Mis an essential submodule of E, then λ∈J=J(Λ) by a theorem of Utumi (e.g. [F2, p. 76, Theorem 19.27(a)]) contradicting the J= 0 assumption, and completing the proof. 8.3. Proposition. If Sis any semisimple right R-module with injective hull E=E(S), then the endomorphism ring Λhas radical (1) J(Λ) = {λ∈∧ker λ⊇S}, 208 C. Faith and moreover, (2) J(Λ) = annΛS. Furthermore, (3) Λ=Λ/J(Λ) = End SR is a full product =Π i∈ALiof full linear rings, where Li= End WDi, and Wiis a vector space over a sfield Di,∀i∈A. Proof: By Utumi’s theorem cited above (proof of 8.2), (2) has the description (1) above. Since a submodule Mof E=E(S) is essential iff M⊇S, this shows that (2) holds. Furthermore since Eis injective, any element of End SRis induced by some λ∈Λ,so (2) ⇒(3). Finally, ¯ Λis a product as described by classical ring theory. 8.4. Corollary. If Eis a minimal injective cogenerator of modR, and Λ = End ER, then ¯ Λ=Λ/J(Λ) is product Πi∈ADiof sfields Di= End(Vi)R, one for each isomorphy class [Vi]of simple modules. Consequently, ¯ Λis a V-ring. Proof: Follows from 8.3. ¯ Λ is thus abelian VNR(=strongly regular), hence is a right and left V-ring. 8.5. Corollary. If (in Theorem 8.3) Eis a minimal injective cogenerator, then E=E(S), where S=⊕Vi, exactly one simple module Viof each isomorphy class, and ¯ Λ=Λ/J(Λ)=Π i∈ADi where Di= End Vi, one for each Vi. Furthermore, ¯ Λis a right and left V-ring. Finally, Λis a right (left) max ring iff J(Λ) is left (right) vanishing. Moreover, Λis right max iff ERsatisfies the acc on kernels of finite products {jn···j2j1}of elements of J(Λ). Proof: Follows from Corollary 8.4, the Harada-Ishii theorem, and the Second Max Theorem. Rings whose modules have maximal submodules 209 8.6. Corollary. If the minimal injective cogenerator Eof mod-Rsatisfies the acc on essential submodules (equivalently, E/soc Eis Noetherian), then Λ = End ERis a right max ring. Proof: Since Λ/J(Λ) is a V-ring (both sides) hence a max ring, then by Hamsher’s theorem, Λ is right max iff J(Λ) is left vanishing. But this follows from Corollary 8.5 and the Harada-Ishi Theorem as in the proof of Theorem 6. (Since soc Eis the intersection of all essential submodule by a theorem of Kasch-Sandomierski, the parenthetical equivalence holds.) Remark 8.6A. The condition of Corollary 8.6 implies that E(V)is Noetherian for any simple module V, and by Corollary 1.1, this is also a sufficient condition for Rto be right max. 8.7. Theorem (Partial Converse of Theorem 6).If Eis an injective cogenerator for mod-R, and if Λ = End ERhas essential left socle then Eis a Bass module. Proof: The proof is a straightforward application of the Harada-Ishii theorem. For if Mis a proper submodule of E, the fact that Eis an injective cogenerator yields hom(E/M,E)= 0, hence some λ∈Λ with ker λ⊇M. Then, if Λλ0is a minimal left ideal of Λ contained in Λλ, by the Harada-Ishii theorem, E0=kerλ0is a maximal submodule containing ker λ, hence M. In the proof of the next theorem, we let ker L=∩λ∈Lker λ. 8.8. Theorem. For a ring R, right injective cogenerator E, and Λ = End ERthe f.a.e.c.’s: (1) Ris right max. (2) Eis a Hamsher module. (3) Λ/L has nonzero socle for any left ideal L= annΛM, where M is a nonzero submodule of E. Proof: (1) ⇔(2) by Theorem 1. (2) ⇒(3). By the DAC Theorem 6.2, if L= annΛM, then M=kerL, hence, since Eis Hamsher module, M has a maximal submodule M0. Since homR(M/M0,E)= 0 and Eis injective, then there exists λ0∈Λ such that λ0M0= 0 and λ0M=0. Moreover, if L0= annΛM0, then by the DAC Theorem 7, annEL0= M0, and since M∩(ker λ0)=M0, then: annE(L+Λλ0) = (ker L)∩(ker λ0)=M∩(ker λ0)=M0= annEL0.