RINGS WHOSE MODULES HAVE MAXIMAL SUBMODULES

A ring R is 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 E of 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 E in the sense that rad E = 0; equivalently, there is an ordinal α such that rad(E(V )) = 0 for each simple module V . This holds iff each rad(E(V )) has a maximal submodule, or is zero (Theorem 2). If follows that R is 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 R via the endomorphism ring Λ of any injective cogenerator E of mod-R; namely, Λ/L has a minimal submodule for any left ideal L = annΛ M for a submodule (or subset) M = 0 of E (Theorem 8.8). Then Λ/L0 has 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 = annΛ M as above (e.g. as in (2)) and L ′ finitely generated (Corollary 8.9A). 1Hamsher modules are called max modules by Shock [S].


One-Module Theorem.
A ring R is 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 V is a Hamsher module.
Proof: A module E cogenerates the category mod-R of 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 E is 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 V range over all simple R-modules.Then E is a cogenerator module for mod-R ( [F1, p. 167, prop. 3.55]).Let P V be the projection E → E(V ).Then, in the above, 0 = M = h(M ) ⊆ E implies 0 = P V h(M ) = M V ⊆ E(V ) is a nonzero submodule of E(V ) for some V , and so M has a maximal submodule, as before, whenever E(V ) is a Hamsher module for all V .
Note.E(V ) is direct summand of any cogenerator E of mod-R, hence the Hamsher condition on E(V ) is a consequence of that on E in Theorem 1.Moreover, this is sufficient for E to be Hamsher.

Corollary.
If R is a ring such that each simple module V has Noetherian injective hull E(V ), then R is 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: R is right V -ring in case R has the equivalent properties.(See [F1, p. 356, 7.32A [B], [H]) if for every sequence {a n } ∞ n=1 of elements of A, there is an n ≥ 1 so that

Kaplansky's Theorem
Expressed otherwise: R is a max ring iff R/J is a V -ring, and J is vanishing.The radical series rad α (M ) is defined inductively for each ordinal α in the usual way, where rad (M ) is the intersection of all maximal submodules of M, rad α+1 (M ) = rad(rad α (M )) for any ordinal, and rad for each limit ordinal β.

Second Max Theorem ([H], [K]
).A ring R is right max iff R/J is right max and J is left vanishing.
We next show that the modules in the radical series are test submodules for a Hamsher module. 3The f.a.e.c.'s on a right R-module M .
Proof: (1) ⇒ (2) is obvious, and (2) ⇒ (3) follows by cardinal number theory for any α of cardinal greater than that of R. (3) ⇒ (1).If S = 0 is a submodule of M , then S rad λ (M ) for least ordinal λ < α, and obviously λ is not a limit ordinal, so S ⊆ rad λ−1 (M ).If S = rad λ−1 M , then S has a maximal submodule since rad S = rad λ (M ) = S.And if S = rad λ−1 (M ), then S is not contained in a maximal submodule M of rad λ−1 (M ), hence S ∩ M is a maximal submodule of S. This proves that M is a Hamsher module.

Corollary.
Let E be a right cogenerator module for R. The R is right max iff E has transfinite nilpotent radical.A n.a.s.c. for C. Faith this is that E(V ) have transfinite nilpotent radical for each simple right R-module V .

Lemma.
If M is a quasi-injective right R-module, then so is every fully invariant submodule, in particular, so is rad α (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 and since E is injective, then every element λ 0 ∈ Λ 0 = End E 0 is induced by an element λ ∈ Λ.Since λ induces an endomorphism λ in M , and since λ(M 0 ) ⊆ M 0 by the hypothesis that M 0 is fully invariant in M , then It follows that rad α+1 (M ) is quasi-injective for all α, since rad α+1 (M ) is fully invariant in rad α (M ) which by an inductive hypothesis may be assumed to be quasi-injective.Furthermore, rad β (M ) is fully invariant hence quasi-injective for each limit ordinal β, since it is the intersection of fully invariant submodules of M .

Theorem.
For a ring R, the f.a.e.c.'s: (1) R is right max.

Corollary. If a right R module M is faithful and has transfinite nilpotent radical, then R has transfinite nilpotent radical J.
Proof: One shows inductively that rad α (M ) ⊇ MJ α , where J = rad R.

Note.
Let R be a commutative Noetherian ring.Then J ω = 0 by the Krull intersection Theorem and if R is a domain, then I ω = 0 for any ideal I = R ([Z-S, p. 216, Theorem 12 and Corollary]).Thus, J is transfinite but not T -nilpotent when R is e.g., a Noetherian local domain not a field.

LOEWY SERIES AND TRANSFINITE SEMISIMPLE MODULES
A descending or dual Loewy series for a module M is descending chain {M α } α∈Λ of submodules indexed by an ordinal Λ such that M 0 = M , and M α /M α+1 is semisimple for any limit ordinal β ∈ Λ.We say that M is transfinitely semisimple if there is a descending Loewy series {M α } with M α = 0 for some α ∈ Λ.

Theorem. Any transfinitely semisimple module M is a Hamsher module.
Proof: By transfinite induction, for each M α as defined above, hence rad α (M ) = 0 for some ordinal α, and Theorem 1 applies: M is Hamsher module.
By Theorem 1, we also have the following: In the next corollary, we see what happens to Λ when E is Noetherian.

Corollary.
If E is a Noetherian quasi-injective right module over R, then Λ = End E R is a right perfect ring, hence a right max ring.
Proof: By the Harada-Ishii DAC cited in the proof of Theorem 6, E R 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 F satisfies the double annihilator conditions (DAC) I = ann R ann F I (see, e.g.[F1]).We next prove another DAC for F .
7. Dac Theorem. 4 If F is any right cogenerator of R, and I and M are submodules of R R and F R respectively, then they satisfy the DAC's: where Ω = End F R .

Proof:
(1) Since F is a cogenerator then R/I → F α for some cardinal α, and if (x i ) is the image in F of the coset 1 + I in R/I, one sees that I = ann R {x i }, so (a) follows.
(2) F/M embeds in a direct product F α of copies of F , and hence there is a map h : F → F α that has ker h = M .Then, if p α : Then, (4) where L = Σ α Ωω α .

INJECTIVE COGENERATORS
If any cogenerator of mod-R is a Hamsher module, then R is a right max ring.In this section we list two conditions on a minimal injective cogenerator E that are each necessary and sufficient in order that R be a right V -ring: (1) rad E = 0. (Theorem 8.1) and (2) E R is a Bass module, and Λ = End E R has zero Jacobson radical (Theorem 8.2).

Theorem.
Let E be a minimal injective cogenerator of R, and W the direct sum of a complete set of non-isomorphic simple right Rmodules.(Thus, E is the injective hull of W , and W is the socle of E.) Then, the f.a.e.c.'s: (1) R is a right V -ring.
(2) ⇒ (1).If V is a simple submodule of E, then (2) implies that there exists a maximal submodule M of E not containing V .Then since V ∩ M = 0, and

Theorem. If the right minimal injective cogenerator E of a ring R is a Bass Module, and if Λ = End E R has zero Jacobson radical, then R is a right V -ring (and E is semisimple).
Proof: Let W = soc E, the sum of all simple module, one for each isomorphy class.If W = E, then every submodule of E is a direct summand, hence is injective, so R is right V -ring.We may therefore assume that E = W , and hence by our Bass module assumption that there is a maximal submodule M of E that contains W . Since V = E/M → W , there is an endomorphism λ of E such that ker λ = M .Since M is 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.

Proposition. If S is any semisimple right R-module with injective hull E = E(S), then the endomorphism ring Λ has radical
(1) and moreover, (2) Furthermore, is a full product = Π i∈A L i of full linear rings, where L i = End W Di , and W i is a vector space over a sfield D i , ∀i ∈ A.
Proof: By Utumi's theorem cited above (proof of 8.2), ( 2) has the description (1) above.Since a submodule M of E = E(S) is essential iff M ⊇ S, this shows that (2) holds.Furthermore since E is injective, any element of End S R is induced by some λ ∈ Λ, so (2) ⇒ (3).Finally, Λ is a product as described by classical ring theory.

Corollary. If E is a minimal injective cogenerator of mod-R, and
Proof: Follows from 8.3.Λ is thus abelian V NR (=strongly regular), hence is a right and left V -ring.

Corollary. If (in Theorem 8.
3) E is a minimal injective cogenerator, then E = E (S), where S = ⊕V i , exactly one simple module V i of each isomorphy class, and Furthermore, Λ is a right and left V -ring.Finally, Λ is a right (left) max ring iff J(Λ) is left (right) vanishing.Moreover, Λ is right max iff E R satisfies the acc on kernels of finite products {j n • • • j 2 j 1 } of elements of J(Λ).
Proof: Follows from Corollary 8.4, the Harada-Ishii theorem, and the Second Max Theorem.

Corollary.
If the minimal injective cogenerator E of mod-R satisfies the acc on essential submodules (equivalently, E/ soc E is Noetherian), then Λ = End E R is 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 E is 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 R to be right max.

Theorem (Partial Converse of Theorem 6)
. If E is an injective cogenerator for mod-R, and if Λ = End E R has essential left socle then E is a Bass module.
Proof: The proof is a straightforward application of the Harada-Ishii theorem.For if M is a proper submodule of E, the fact that E is an injective cogenerator yields hom(E/M, E) = 0, hence some λ ∈ Λ with ker λ ⊇ M .Then, if Λλ 0 is a minimal left ideal of Λ contained in Λλ, by the Harada-Ishii theorem, E 0 = ker λ 0 is a maximal submodule containing ker λ, hence M .
In the proof of the next theorem, we let ker L = ∩ λ∈L ker λ. 8.8.Theorem.For a ring R, right injective cogenerator E, and Λ = End E R the f.a.e.c.'s: (1) R is right max.
(3) Λ/L has nonzero socle for any left ideal L = ann Λ M , where M is a nonzero submodule of E.

8.9B. Corollary. If E is an injective cogenerator of mod-R with left
Loewy (equivalently, left semiartinian) endomorphism ring Λ, then R is right max and Λ is right perfect.Moreover, R has just finitely many simple right modules.
Proof: If Λ is left Loewy, then Λ/L has nonzero socle for all left ideals L = Λ, so Theorem 8.8 applies to establish that R is right max.Since Λ = Λ/J(Λ) is also left Loewy and right self-injective (see, e.g.(3) of Prop.8.3), then Λ is semisimple Artinian and J = J(Λ) is left vanishing, hence Λ is right perfect.(See, for example, the discussion in [C-P, esp.Lemma 1 and the proof of Proposition 2].) Furthermore, since Λ is semisimple and isomorphic to the endomorphism ring of the socle S of E (see the proof of 8.3), then S has finite length.This shows that the isomorphy set of simple right R-modules is finite.8.10.Corollary.If E is an injective cogenerator of mod-R, and Λ = End E R , then R is right max iff J = rad R left vanishing, and Λ/L has nonzero left socle for any left ideal L = ann Λ M , where M is a nonzero R-submodule of E annihilated by J.
Proof: One knows that F = ann E J is an injective cogenerator of mod-R/J (F is injective as an R/J-module and contains a copy of each simple R-module).Moreover, F is a fully invariant R-submodule of E, hence, by injectivity of E, The corollary now follows from Hamsher's Second Theorem and Theorem 8.8.
is left vanishing and (0, M) 2 = 0, and then an easy computation shows that J(R) is left vanishing.

REMARKS ON THE LITERATURE
A module M is quotient finite dimensional (= q.f.d.) provided that all factor modules have finite Goldie dimension, i.e., contain no infinite direct sums.Generalizing a theorem of Shock [S], Camillo [C1] proved that an R-module M is q.f.d.iff every submodule N contains a finitely generated submodule K with N/K having no maximal submodules.This implies that a q.f.d.module M is Noetherian iff every factor module M/K is Hamsher.Since linearly compact modules are q.f.d., then by duality theory [M] F2]).The example of a right but not left V -ring R of the author's in [F4] is a V NR of left Loewy length 2 hence left max.
As an application of Theorem 1, we prove in [F3] that for a commutative ring R that the f.e.c.'s : (1) R is locally a perfect ring (= R m is perfect at each maximal ideal m); (2) R m is a max ring for each maximal ideal m; (3) R is a max ring.

QUESTIONS
(1) If Λ = End E R is a right max ring, for a minimal injective cogenerator E of mod-R, is R right max?
(2) If R is right max, is Λ?
In [C2], Camillo proves that a right max right and left P ID R is simple, and that given two maximal right ideals, pR and qR, either R/pqR or R/qpR is semisimple.
(3) Characterize when a P ID ring R is right (or left) max.It is of course if R/aR (or R/Ra) is semisimple for any 0 = a ∈ R. (See [C2].) (4) (Hamsher [H]) When is a full linear ring right or left max? (Regarding the corresponding question for V -rings, see Osofsky [0].) ) rad M = 0 for each right R-module M .(V3) Every right ideal I = R is 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 = M for every M = 0.
6. Theorem.Let E be an quasi-injective right R-module that contains a copy of each simple image of E and Λ = End E R .If E is a Bass module, then Λ has essential left socle, soc Λ. Proof: By the Harada-Ishii ([H-I]) double annihilator condition (= DAC) for a quasi-injective modules, ann Λ ann E I = I for finitely generated left ideals of Λ, one can show that each such I = 0 contains a minimal left ideal

8. 11 .
Example.Let M be any bimodule over a right max ring A. Then the split-null or trivial extension R = (A, M ) is a right max ring.Proof: Let J(A) be the (left vanishing) radical of A. Then J one shows that a Morita ring R(= R has a Morita duality) is right max iff left Loewy (= semi-Artinian and iff R is right and left Artinian.Results of Camillo and Fuller [C-F1], [C-F2] and Nastasescu and Popescu [N-P] are germane here: A left Loewy ring R of finite Loewy length is right max ([C-F1], [N-P]).More generally, any left Loewy ring with acc on primitive ideals is right max ([C-