ON CLASSES OF MODULES CLOSED UNDER INJECTIVE HULLS AND ARTINIAN PRINCIPAL IDEAL RINGS

In this work we consider some classes of modules closed under certain closure properties such as being closed under taking submodules, quotients, injective hulls and direct sums. We obtain some characterizations of artinian principal ideal rings using properties of big lattices of module classes. Mathematics Subject Classification (2010): 16D50, 16D80


Introduction
In this work we consider some classes of modules closed under certain closure properties such as being closed under taking submodules, quotients, injective hulls and direct sums.We use the notation L {≤} , L { } , L {E} and L {⊕} describe as follows.We will denote L {≤} the class of hereditary classes, L { } the class of module classes closed under taking quotients, L {E} the class of module classes closed under taking injective hulls, L {⊕} the class of module classes closed under taking direct sums.L {≤,E} will denote the class of module classes closed under taking submodules and injective hulls.In general, if A is a set of closure properties, we denote by L A the class of module classes closed with respect to the closure properties in A. If A denotes a subset of {≤, , E, ⊕} we should notice that L A becomes a big lattice ordered by class inclusion with infima given by intersections.
There are many lattices of module classes of this type which are interesting to study for themselves.In this paper we will study lattices of module classes like L {≤,E} , L { ,E} .
We obtain some characterizations of artinian principal ideal rings using properties of big lattices of module classes.In the sequel, R denotes an associative ring with identity.Definition 2.1.Let A be a set of closure properties and let C be a class of Rmodules, we denote by ξ A (C ) the least class of modules containing C and being closed under the properties in A.
We omit the easy verification of the following proposition. ( Proof.By definition, ξ P (C ) is the least class in L P containing C , as Now we describe generation in the lattices L {≤,E} , L {≤,⊕} and L { ,E} .
Proof.Let N ∈ ξ ≤ (C ) then there exists a monomorphism t : N C with C ∈ C .We obtain a commutative diagram: The next result is an immediate consequence of Proposition 2.4.
Remark 2.8.Notice that L {≤,E} is a complete and distributive big lattice, where infima and suprema are given by intersection and union of classes respectively.Furthermore R-mod and {0} are the greatest and least elements of the lattice.
Proof.Clearly C ⊆ ξ ≤ ξ ⊕ (C ).Now we will prove that ξ ≤ ξ ⊕ (C ) is a class closed under taking submodules and direct sums.As C clearly is a hereditary class, it suffices to show that it is closed under taking direct sums.Let {M i } i∈I ⊆ ξ ≤ ξ ⊕ (C ) be a family, then for every M i there exists a family {C ij } j∈Ji of modules in C such that there is a monomorphism Remark 2.12.Observe that for each n ∈ N and for each class C of R-modules, We prove by induction that η n (C ) ⊆ D for all n ∈ N.For n = 0 we have closed under taking injective hulls and quotients.Therefore η n+1 (C ) ⊆ D and thus Proof.An immediate application of push outs and pull backs.
(2) For each left injective R-module I, if there exists an epimorphism I K, then there exists a monomorphism K I.
Thus there exists a monomorphism K I. ( By hypothesis, there exists a monomorphism Theorem 3.3.For a ring R, the following conditions are equivalent: (1) L {≤,E} = L { ,E} .
(2) For each left injective R-module I we have: There exists a monomorphism K I if and only if there exists an epimorphism I K. Proof.
(1) ⇒ (2) From Theorem 3.1 we have that, if there exists an epimorphism I K then there also exists a monomorphism K I. Now if there exists a monomorphism K I we can change I for E(K).If L ∈ ξ (E(K)), then there exists an epimorphism E(K) L. Let E(L) be the injective hull of L. As we pointed out at the beginning then there exists a monomorphism L E(K) which extends to Proof.We will show that each projective module is injective, a condition which is equivalent to R being quasi-Frobenius by the Faith-Walker Theorem (see [2], [6] or [8]).Let P be a projective R-module.As ξ ,E (E(P )) ∈ L {≤} then P ∈ ξ ,E (E(P )).Let n ∈ N be least such that P ∈ η n (E(P )).Note that if n = 0 then P ∈ η 0 (E(P )) = {E(P )} and therefore P is an injective R-module.
Therefore P is injective.Proof.Let 0 = C ∈ C ⊆ ξ ≤,E (I), this implies that there exists a monomorphism C I so that E(C) is a direct summand of I, as I is indecomposable then Theorem 3.7.Let R be a left noetherian ring.Then the following assertions are equivalent for a class C of left R-modules: (1) C is an atom in L {≤,E} .

Artinian principal ideal rings
The following theorem contains some well known results about artinian principal ideal rings (i.e.left and right artinian, left and right principal ideal rings), we include them here for convenience.
Theorem 4.1.The following properties are equivalent: (1) R is an artinian principal ideal ring.
(2) R is a left principal ideal ring and a quasi-Frobenius ring.
(3) The injective hull and the projective cover of each (right or left) finitely generated R-module are isomorphic.
(4) For each left R-module M , R soc(M ) ∼ = M JM and for each right R-module N , soc(N ) R ∼ = N N J , where J denotes the Jacobson radical of the ring R. (5) For each ideal I of R, R/I is quasi-Frobenius.
(2) R is an artinian principal ideal ring.
Notice that condition (5), which we are assuming, holds also for R I -modules for each two-sided ideal I. Thus we conclude that R I is quasi-Frobenius for each two-sided ideal I. Now we use Theorem 4.1.
(2) ⇒ (4) and ( 2 (4) ⇒ (2) We claim that ξ ≤ ({F ∈ R-mod|F is free })=R-mod.Indeed, as each R-module M is a quotient of a free left R-module F and as F ∈ ξ ≤ ({F ∈ R-mod|F is free }) ∈ L then M belongs to this class.Thus each R-module R M is a submodule of a free module and this is equivalent to R being a quasi-Frobenius ring (see [7]).Thus R is a quasi-Frobenius ring.Observe that condition (4) holds also for R I for each two-sided ideal I. Thus R I is a quasi-Frobenius ring for each two-sided ideal I. Now use Theorem 4.1.
Recall that artinian modules are precisely the left R-modules such that all of their quotients are finitely cogenerated.As the class of finitely cogenerated left modules is closed under taking submodules and injective hulls, we have the following result.(1) M is artinian.
Recall that a ring R is called left co-noetherian if the injective hull of each simple R-module is artinian [11].Proof.We show that each non zero R-module contains a simple submodule.If M is a non zero R-module, then the class ξ ≤ (E(M )) belongs to L {≤,E} .As a consequence, a simple quotient S of a non zero cyclic submodule of M , also embeds in E(M ).Thus, such a simple quotient S embeds in M , inasmuch as M is an essential submodule of E(M ).
In the proof of the following Theorem we adapt an idea of [5] which also uses a Lemma in [10].Recall that a ring is left local when all of its simple left modules are isomorphic.In particular for j = i, we obtain Hom R ( Ii+1 Ii , E) ∼ = Li Li+1 = 0.As E is an injective cogenerator of R-mod, then Ii+1 Ii = 0. From this we obtain I i+k = I i for all k ≥ 0. Therefore R satisfies the ascending chain condition for two-sided ideals.
In [4] Bronowitz and Teply proved that the rings for which all of its hereditary torsion theories are cohereditary are precisely the finite products of left local right perfect rings.

Remark 2 . 9 .
If C is a class of R-modules, then ξ ≤ ξ ⊕ (C ) = {M | there exists a monomorphism M i∈I C i , with I a set, and a class of modules closed under taking submodules and direct sums containing C .If M ∈ ξ ≤ ξ ⊕ (C ), then there exists a monomorphism M i∈I C i with {C i } i∈I ⊆ C .Hence {C i } i∈I ⊆ D, then by hypothesis i∈I C i ∈ D, thus M ∈ D. Therefore ξ ≤ ξ ⊕ (C ) ⊆ D. Definition 2.11.Let η = ξ E ξ we define η 0 = Id, η n+1 = ηη n for all n ∈ N and for a class of R-modules C we define η ∞ (C ) = ∪ n∈N η n (C ).
is the least class of the classes closed under taking quotients and injective hulls containing C .Consider D ∈ L { ,E} such that C ⊆ D.
) ⇒ (1) Let us take C ∈ L {≤,E} , it suffices to prove that C is closed under taking quotients.If M ∈ C and f : M N is an epimorphism, consider the following commutative diagram:

Theorem 3 . 2 .
L { ,E} ⊆ L {≤,E} if the following condition holds for each injective left R-module I: If there exists a monomorphism K I then there exists an epimorphism I K. Proof.Let C ∈ L { ,E} , it suffices to prove that C is closed under taking submodules.Let us take M ∈ C and let N M be a monomorphism, then there exists an epimorphism f : E(M ) N .Then N ∈ C since C is closed under taking quotients and injective hulls.

Theorem 3 . 5 .Proposition 3 . 6 .
If L {≤,⊕} ⊆ L {E} , then R is a left V -ring and a left noetherian ring.Proof.Let C = {M | M is semisimple}.Clearly C is a class of R-modules closedunder taking submodules and direct sums, then C is closed under taking injective hulls.Hence E(M ) is semisimple for each M ∈ C , this implies that M is a direct summand of its injective hull, so that M = E(M ).Thus every semisimple module is injective, therefore R is a left V -ring.We also have that i∈N E(S i ) = i∈N S i is semisimple and injective, therefore R is left noetherian.If I is an indecomposable injective left R-module, then ξ ≤,E (I) is an atom in L {≤,E} .

( 2 ) ⇒ ( 1 )( 1 ) ⇒ ( 2 )
It follows from Proposition 3.6.Assume that C is an atom of L {≤,E} and let us take C ∈ C .Then E(C) ∈ C .As R is a left noetherian ring we have that there exists a family {I α } α∈J of left indecomposable injective modules such that ⊕ α∈J I α = E(C).For α ∈ J we have that ξ ≤,E (I α ) ⊆ C , but as C is an atom then ξ ≤,E (I α ) = C .
is left selfinjective.For each left ideal I of R there exists an epimorphism f : R I , therefore I is cyclic.Thus R is a left principal ideal ring in particular it is left noetherian.As R is a left noetherian and left selfinjective ring, R is a quasi-Frobenius ring.Thus condition 2) of Theorem 4.1 is fulfilled.

Theorem 4 . 3 .
If L {≤,E} ⊆ L { ,E} , then for each R-module M the following statements are equivalent:

Proposition 4 . 4 .Proposition 4 . 6 .
If L {≤,E} ⊆ L { ,E} , then R is a left co-noetherian ring.Proof.If S is a simple left R-module, then S is finitely cogenerated, then also E(S)is finitely cogenerated.By Theorem 4.3 we have that E(S) is artinian, therefore R is co-noetherian.Proposition 4.5.If L {≤,E} ⊆ L { ,E} , then E(R) is a cogenerator of R-mod.Proof.Let S a simple R-module.If 0 = x ∈ S then Rx = S.As we have an epimorphism R / / / / Rx and ξ ≤,E (R) ∈ L {≤,E} ⊆ L { ,E} , then by Theorem 3.1 there exists a monomorphism Rx E(R).Therefore E(R) contains a copy of each simple left R-module, thus E(R) is an injective cogenerator.If L {≤,E} ⊆ L { ,E} , then R is a left semiartinian ring.

Theorem 4 . 7 .
If R is a left local and left co-noetherian ring, then R satisfies the ascending chain condition for two-sided ideals.Proof.Since R is a left local ring then there exists just one simple left R-module S up to isomorphism.Then E = E(S) is a cogenerator for R-mod.Also notice that it is left artinian by the actual hypothesis.Let I 0 = 0 ⊆ I 1 ⊆ . . .be an ascending chain of two-sided ideals of R. Taking L i = {x ∈ E|I i x = 0} we obtain a descending chain of left submodules of E, L 0 = E ⊇ L 1 ⊇ L 2 ⊇ . . . .As E is left artinian then there exists i ∈ N such that L i+k = L i for all k ≥ 0. We may identify L j with Hom R R Ij , E by letting correspond x in L j to the homomorphism sending 1 + I j into xsequence of left R-modules and as E is a left injective R-modules, we obtain an exact sequence of abelian groups0 −→ Hom R R I j+1 , E −→ Hom R R I j , E −→ Hom R I j+1 I j , E −→ 0.Then we get the exact sequence 0 −→ L j+1 −→ L j −→ Hom R I j+1 I j , E −→ 0.