STABLE TORSION THEORIES AND THE INJECTIVE HULLS OF SIMPLE MODULES

A torsion theoretical characterization of left Noetherian rings over which injective hulls of simple left modules are locally Artinian is given. Sufficient conditions for a left Noetherian ring to satisfy this finiteness condition are obtained in terms of torsion theories. Mathematics Subject Classification (2010): 16S90, 13E05


Introduction
A famous open problem in ring theory is Jacobson's conjecture which asks, for a two sided Noetherian ring R with Jacobson radical J, whether it is true that ∞ i=1 J i = 0. Jategaonkar [11] showed in 1974 that fully bounded Noetherian (FBN for short) rings satisfy Jacobson's conjecture.A key step in his proof is that over an FBN ring R, finitely generated essential extensions of simple left modules are Artinian.This is equivalent to the property ( ) injective hulls of simple left R-modules are locally Artinian.
The natural question whether this holds for arbitrary Noetherian rings was shown not to be true by Musson (see [14] or [15]).
It should be noted that if R is a Noetherian ring which satisfies ( ), then R satisfies the Jacobson's conjecture.This makes the property ( ) interesting on its own and some Noetherian rings have been tested whether they satisfy this property or not.Recently, interest in this property has increased when Carvalho, Lomp, and Pusat-Yılmaz [2] considered this property for Noetherian down-up algebras and started the characterization of these algebras with property ( ) by giving a partial answer.This characterization has been then completed in the following works of Carvalho and Musson, and Musson [3], [16].These results have been followed by a characterization of finite dimensional solvable Lie superalgebras g over an algebraically closed field of characteristic zero whose enveloping algebra U (g) satisfies ( ) [10].Most recently, a complete characterization of Ore extensions with property ( ) has been obtained in [1].
There is a way in which torsion theories and property ( ) can be linked.We prove in Proposition 2.1 below that for a left Noetherian ring R, injective hulls of simple left R-modules are locally Artinian if and only if Dickson's torsion theory is stable.This connection makes it possible to carry the study of Noetherian rings satisfying property ( ) to the area of stable torsion theories.Using the results from stable torsion theories, we are able to obtain more examples of rings which satisfy property ( ).
The organization of the paper is as follows.In the first section we briefly recall the general notions of torsion theories.We then consider in the next section stable torsion theories and obtain certain conditions on rings which lead to property ( ).
The next part of the paper is devoted to some classes of rings that satisfy these certain conditions.

Generalities on torsion theories
Let R be an arbitrary associative ring with unity and let R-mod denote the category of left R-modules.Dickson had the idea of carrying the notion of torsion in abelian groups to abelian categories, and he defined in [5] a torsion theory τ on R-mod to be a pair τ = (T τ , F τ ) of classes of left R-modules, satisfying the following properties: We call T τ the class of τ -torsion modules and F τ the class of τ -torsionfree modules.
Let A and B be nonempty classes of left R-modules.
, then B is said to be the right orthogonal complement of A. We say that the pair (A, B) is a complementary pair whenever A is the left orthogonal complement of B and B is the right orthogonal complement of A. In particular, if τ is a torsion theory, then (T τ , F τ ) is a complementary pair and every such pair defines a torsion theory.
An immediate consequence of the definition is that the class of torsion modules for a torsion theory τ is closed under extensions.Similarly, the class of torsionfree

STABLE TORSION THEORIES AND THE INJECTIVE HULLS OF SIMPLE MODULES 91
modules is closed under extensions too.While it is not required in the definition of a torsion theory, we will be working with torsion theories such that the class of torsion modules is closed under submodules.Such torsion theories are called hereditary.
A nonempty set L of left ideals of a ring R which satisfies the following conditions is called a Gabriel filter :

Stable torsion theories
A hereditary torsion theory τ on R-mod is called stable if its torsion class is closed under injective hulls.One of the equivalent conditions for D to be stable is that modules with essential socle are D-torsion [5, 4.13].
Dickson characterized those rings for which Dickson's torsion theory in the category R-Mod is stable.Indeed he considered property ( ) for any abelian category with injective envelopes.Translating his results to the language of the present paper, for a left Noetherian ring we obtain a connection between the stability of Dickson's torsion theory and property ( ) in the following, which is the main result of the paper.Note that if N is an essential submodule of a module M , then we denote this fact as N ≤ e M .Proposition 2.1.The following are equivalent for a left Noetherian ring R.
(i) R satisfies property ( ); (ii) Dickson's torsion theory is stable; (iii) Any D-torsion R-module can be embedded in a D-torsion injective R-module; (iv) Any injective R-module A decomposes as A = A t ⊕ F , where A t is the D-torsion part of A and F is unique up to isomorphism and has no socle; (v) If A is an essential extension of its socle, then it is D-torsion; (vi) For any left R-module A, its torsion part A t is the unique maximal essential extension in A of its socle soc(A).The equivalence of (ii -vi) follows from [5, 4.13].
Hence, over a Noetherian ring, the stability of Dickson's torsion theory is a necessary and sufficient condition for property ( ).We will be looking for cases in which Dickson's torsion theory is stable for a left Noetherian ring R.There are two such cases which imply the stability of Dickson's torsion theory, as we will see shortly.
For a ring R we denote the family of all hereditary torsion theories defined on R-mod by R-tors, and this corresponds bijectively to a set [8,Proposition 4.6].
Recall that a partial order can be defined on R-tors by setting τ ≤ σ if and only if every τ -torsion left R-module is σ-torsion, or equivalently every σ-torsionfree left R-module is τ -torsionfree [8, Proposition 2.1].
For example, with respect to this ordering, Goldie's torsion theory is the smallest torsion theory in which every cyclic singular left R-module is torsion and Dickson's torsion theory is the smallest torsion theory in which every simple left R-module is torsion.

Cyclic singular modules with nonzero socle
We now give a sufficient condition for a torsion theory to be stable.The following result is present in the proof of Proposition 1 in [21] but it is not given explicitly.
We record it as a lemma and give its proof for the convenience of the reader.Lemma 3.1.Any generalization of Goldie's torsion theory G is stable.
This research was supported by Fundação para a Ciência e a Tecnologia -FCT by the grant SFRH/BD/33696/2009.
(i) I ∈ L and a ∈ R implies ann R (a + I) belongs to L.(ii) If I is a left ideal and J ∈ L is such that ann R (a + I) ∈ L for all a ∈ J, then I ∈ L.Hereditary torsion theories in R-mod and Gabriel filters in R are in one-to-one correspondence[19, Theorem VI.5.1].1.1.Goldie's torsion theory.Let M be a left R-module.An element m ∈ M is called a singular element of M if ann R (m) = {r ∈ R | rm = 0} is an essential left ideal of R. The collection Z(M ) of all singular elements of M is a submodule of M called the singular submodule of M .A module M is called singular if Z(M ) = M and it is called nonsingular if Z(M ) = 0.The class F G of all nonsingular leftR-modules forms a torsionfree class for a hereditary torsion theory on mod-R.We call this Goldie's torsion theory and denote it by G [9,20].For any right R-module M , its Goldie torsion submodule is t G (M ) = {m ∈ M | m + Z(M ) ∈ Z(M/Z(M ))}.The Gabriel filter corresponding to Goldie's torsion theory is the set of all essential left ideals L of R such that there exists an essential left ideal L of R such that for every x ∈ L , ann R (x + L) = {r ∈ R | rx ∈ L} is essential in R. Hence, Goldie's torsion class T G is precisely the class of modules with essential singular submodule, and corresponding torsionfree class F G is the class of nonsingular modules.1.2.Generation & cogeneration of torsion theories, Dickson's torsion theory.Let C be a class of left R-modules.If F is a right orthogonal complement of C and T is a left orthogonal complement of F, then the pair (T , F) is a torsion theory in R-mod, called the torsion theory generated by C. If T is the left orthogonal complement of C and F is the right orthogonal complement of T , then the pair (T , F) is a torsion theory, called the torsion theory cogenerated by C. Let S be a representative class of nonisomorphic simple left R-modules.Then the torsion theory D generated by S is called Dickson's torsion theory.The class of D-torsionfree left R-modules are the right orthogonal complements of simple left R-modules while the class of D-torsion left R-modules are the left orthogonal complements of the class of D-torsion modules.In particular, every simple left R-module is D-torsion.Hence the class of all D-torsionfree modules consists of all soclefree left R-modules.Moreover, if M ∈ T D , then M is an essential extension of its socle.A left R-module M is called semi-Artinian if for every submodule N = M , M/N has nonzero socle.The following result is well known in the literature.Lemma 1.1.A left R-module M is D-torsion if and only if it is semi-Artinian.Recall that a module has finite length if and only if it is both Noetherian and Artinian.In fact, we can still have finite length if the module is Noetherian and semi-Artinian.Lemma 1.2.[19, Proposition VIII.2.1]A left R-module M has finite length if and only if it is Noetherian and semi-Artinian.
STABLE TORSION THEORIES AND THE INJECTIVE HULLS OF SIMPLE MODULES 93Proof.(i) ⇒ (ii) We show that ( ) implies the stability of D. Let M be a D-torsion left R-module.Then M has an essential socle and so its injective hull E(M ) is a direct sum of injective hulls of simple left R-modules because R is Noetherian.Then E(M ) is locally Artinian by assumption.We show that E(M ) is D-torsion.Let f : E(M ) → F be an R-module homomorphism, where F is a left R-module with zero socle.For any x ∈ E(M ), since Rx is Artinian, the restriction f : Rx → F is zero.It follows that f is zero and thus E(M ) is also D-torsion by definition.(v) ⇒ (i) Let S be a simple left R-module and E(S) be its injective hull.Let 0 = F ≤ E(S) be a finitely generated submodule of E(S).Then S ≤ e F ≤ e E(S) and having an essential simple submodule, F has an essential socle.This means, by assumption, that F is D-torsion, i.e.F is semi-Artinian by Lemma 1.1.Since it is a finitely generated module over a left Noetherian ring, F is Noetherian as well.By Lemma 1.2 F is Artinian.Thus E(S) is locally Artinian.