Classification of split TTF-triples in module categories

Pedro Nicolás and Manuel Saorín

Abstract. In our work [9], we complete Jans' classification of TTF-triples [8] by giving a precise description of those two-sided ideals of a ring associated to one-sided split TTF-triples in the corresponding module category.

2000 Mathematics Subject Classification. 16D, 16E

Key words and phrases. Idempotent ideal; Torsion pair; Torsion theory; TTF-triple

 

1. Introduction

Since the 1960s torsion theories have played an important role in algebra. They translate to general abelian categories (and so, significantly, to arbitrary module categories) many features of modules over a PID [3], they have been a fundamental tool for developing a general theory of noncommutative localization [10], they have had a great impact in the representation theory of Artin algebras [6, 5, 1], …

In the context of module categories over arbitrary rings, one of the important concepts related to torsion theory is that of TTF-triple. This notion was introduced by J. P. Jans [8], who proved that TTF-triples in the category of modules over an arbitrary ring are in bijection with idempotent two-sided ideals of the ring . He also proved that this bijection restricts to a bijection between the socalled centrally split TTF-triples and the two-sided ideals generated by a single central idempotent. Then, a natural question arises: which are the idempotent ideals corresponding to the TTF-triples which are not centrally split but only one-sided split?

In section 2, we recall the notion of torsion pair and its basic properties. In section 3, we recall the notion of TTF-triple, its basic properties and Jans' parametrization by means of idempotent two-sided ideals. We also recall some deep results of G. Azumaya relating properties of a TTF-triple and properties of the associated idempotent ideal. These results have been crucial for our classification of one-sided split TTF-triples. In section 4, we give a precise description of those two-sided ideals corresponding to the socalled left split TTF-triples according to Jans' parametrization. The analogous description for right split TTF-triples is more complicated and it is explained in section 5: firstly for 'good' rings, and finally for arbitrary rings.

2. Torsion theory: the axiomatic of Dickson

S. E. Dickson introduced torsion theories (also called torsion pairs) in arbitrary abelian categories [3]. If is an abelian category, a pair of classes of objects of is a torsion pair if it satisfies

1. for all in and in .
2. If for all in then is in .
3. If for all in then is in .

is said to be the torsion class, and its objects are the torsion objects. Similarly, is the torsionfree class and its objects are the torsionfree objects. A torsion pair can be uniquely determined in different ways. For instance, it is uniquely determined by its torsion class, since agrees with the class of objects such that for all torsion objects . Also, torsion pairs in are in bijection with the (isomorphism classes of) idempotent radicals, i.e. subfunctors of the identity functor such that and for each object of . Given a torsion pair in , its associated idempotent radical is uniquely determined by the fact that for each object of the object is the largest torsion subobject of . We say that a torsion pair with idempotent radical is split if is a direct summand of for each object of .

3. TTF-triples

3.1. Jans' classification. Shortly after the axiomatic of S. E. Dickson appeared, J. P. Jans introduced and studied in [8] what he called torsion torsionfree(=TTF) theories (also called TTF-triples) in module categories. The definition still make sense for arbitrary abelian categories, and it is as follows. A triple of classes of objects of an abelian category is a TTF-triple if both and are torsion pairs. A TTF-triple is uniquely determined by its central class, which is said to be a TTF-class since it is both a torsion class of and a torsionfree class of . A TTF-triple is left split (resp. right split) if (resp. ) splits, and it is centrally split if it is both left and right split.

J. P. Jans proved [8, Corollary 2.2], by using a result of P. Gabriel [4], that TTF-triples in the category of right modules over an arbitrary ring are in bijection with the idempotent ideals of that ring . This is the most remarkable result result concerning TTF-triples, and this is why they are interesting mainly in the framework of module categories.

Theorem. Let be a ring. There exists a one-to-one correspondence between:

1. Idempotent two-sided ideals of .
2. TTF-triples in .

The bijection is as follows. Given an idempotent ideal of , the corresponding TTF-triple is the one whose TTF-class is formed by the modules such that . Reciprocally, if is a TTF-triple and is the idempotent radical associated to the torsion class , the corresponding ideal is (where is regarded as a right -module with its regular structure).

J. P. Jans also studied some elementary properties of centrally split TTF-triples and he essentially proved in [8, Theorem 2.4] the following:

Corollary. Let be a ring. The one-to-one correspondence of Theorem 3.1 restricts to a one-to-one correspondence between:

1. (Ideals of generated by single) central idempotents of .
2. Centrally split TTF-triples in .

3.2. The main question. Let be an arbitrary ring, and put , and for the sets of left, centrally and right split TTF-triples in , respectively. The existence of TTF-triples for which only one of the constituent torsion pairs split, which we shall call one-sided split, has been known for a long time [11] (see also the remark 5.2), and so we should have a diagram of the form:

The main question tackled in our work is: What should replace the question marks in the diagram above?

3.3. The work of Azumaya. Some efforts have been made to answer this question (cf. [2], [7], …). Specially useful for us has been the paper of G. Azumaya [2] in which he expresses some deep properties of TTF-triples in module categories in terms of the associated idempotent ideal. We present in the following theorem the results of G. Azumaya we have used [2, Theorem 3, Theorem 6 and Theorem 8].

Theorem. Let be a TTF-triple in a module category and let be the corresponding idempotent ideal of . The following properties hold:

1. is a TTF-class if and only if for some idempotent of .
2. is a TTF-class if and only if for some idempotent of .
3. is closed under submodules if and only if is a flat right -module.
4. is a TTF-class if and only if is closed under submodules and the left -module has a projective cover.

4. Classification of left split TTF-triples over arbitrary rings

Let be an arbitrary ring. Recall that an -module is hereditary -injective if every quotient of a direct sum of copies of is an injective -module. Now we present Theorem 3.1 and Corollary 3.2 of [9].

Theorem. Let be a ring. The one-to-one correspondence of Theorem 3.1 restricts to a one-to-one correspondence between

1. Left split TTF-triples in .
2. Two-sided ideals of of the form where is an idempotent of such that is hereditary -injective as a right -module.

Proof. If the TTF-triple is left split, then is of the form for some idempotent of . The difficult part is to prove that is hereditary -injective as a right -module.

On the other hand, if is an idempotent of such that is a two-sided ideal, then . Then is isomorphic to the triangular matrix ring

where the -bimodule is hereditary -injective in . This property of allows us to prove that is a direct summand of for every -module .

5. Classification of right split TTF-triples

The 'dual' of the Theorem 4 is not true in general but only for some classes of rings.

5.1. Over 'good' rings. Recall that if is a ring, then a -module is hereditary projective (resp. hereditary -projective) in case every submodule of (resp. every submodule of a direct product of copies of ) is projective. Recall also that a -module is called FP-injective if it is injective relative to the class of finitely presented modules, i.e. if vanishes on all the finitely presented -modules. Now we present Proposition 4.5 of [9].

Proposition. Let be a ring and be a left -module. The following conditions are equivalent:

1. For every bimodule structure and every right -module , the right -module is hereditary projective.
2. There exists a bimodule structure such that for every right -module the right -module is hereditary projective.
3. Put . If is the minimal injective cogenerator of , then the right -module is hereditary -projective.
4. The character module is a hereditary -projective right -module.
5. for some idempotent is a hereditary perfect ring and is FP-injective as a left -module.

When is an algebra over a commutative ring , the above assertions are equivalent to:

6. If is a minimal injective cogenerator of , then is a hereditary -projective right -module.

A left -module satisfying the equivalent conditions of the proposition above is said to have a hereditary -projective dual. Now we can formulate the classification of right split TTF-triples under particularly good circumstances (cf. section 4 of [9]):

Theorem. Let be a ring. The one-to-one correspondence of Theorem 3.1 restricts to a one-to-one correspondence between:

1. Right split TTF-triples in whose associated idempotent ideal is finitely generated on the left.
2. Two-sided ideals of of the form where is an idempotent of such that the left -module has hereditary -projective dual.

In particular, when satisfies one of the following two conditions, all the TTF-triples in have the associated idempotent ideal finitely generated on the left:

1. is semiperfect.
2. Every idempotent ideal of which is pure on the left is also finitely generated on the left (e.g.  if is left Nœtherian).

Proof. If is right split, then is hereditary. By Theorem 3.3, is pure as a left ideal. Since is finitely generated on the left, then for some idempotent of . The difficult part is to prove that the left -module has hereditary -projective dual.

On the other hand, let be an ideal like in (2). Since , then is isomorphic to the triangular matrix ring

The fact that the left -module has hereditary -projective dual allows us to prove that the TTF-triple associated to is right split.

We use Theorem 3.3 to prove that if satisfies either condition or then all the TTF-triples in have the associated idempotent ideal finitely generated on the left.

5.2. Over arbitrary rings. The parametrization of the right split TTF-triples over arbitrary rings is more involved. The first observation is the following (cf. Proposition 5.1. of [9]):

Proposition. The one-to-one correspondence of Theorem 3.1 restricts to a one-to-one correspondence between:

1. Right split TTF-triples in .
2. Idempotent ideals of such that, for some idempotent of , one has that and the TTF-triple in associated to is right split.

In the situation of the proposition above one has that , that is, the TTF-triple in associated to has the property that . Therefore the problem of classifying right split TTF-triples reduces to answering the following:

Question. Let be an idempotent ideal of a ring such that , i.e. the where is the associated TTF-triple in . Which conditions on are equivalent to saying that is right split?

The elucidation of these conditions leads us to the following 'arithmetic' definition: Given a right -module and a submodule , we shall say that is I-saturated in when , with , implies that . Equivalently, this occurs when .

When is a subset of , we shall denote by the subset of matrices of with entries in .

Definition. An idempotent ideal of a ring is called right splitting if:

1. it is pure as a left ideal,
2. ,
3. it satisfies one of the following two equivalent conditions:
   1. for every integer and every -saturated right ideal of , there exists such that ,
   2. for every integer and every -saturated submodule of , the quotient is projective as a right -module.

The fact that conditions (i) and (ii) are equivalent is proved in [9, Lemma 5.2].

Finally, we can formulate the general classification of right split TTF-triples (cf. Theorem 5.4 of [9]):

Theorem. Let be an arbitrary ring. The one-to-one correspondence of Theorem 3.1 restricts to a one-to-one correspondence between:

1. Right split TTF-triples in .
2. Idempotent ideals such that for some idempotent of and is a right splitting ideal of with a hereditary perfect ring.

Example. Let be rings, the first one being hereditary perfect, and let be a bimodule such that is faithful. The idempotent ideal

of

is clearly pure on the left and . One can see that is right splitting if, and only if, is FP-injective (equivalently, has a hereditary -projective dual).

Remark. Let be a commutative ring. Denote by , and the sets of left, centrally and right split TTF-triples in , respectively. Then and the last inclusion may be strict. Indeed, since all idempotents in are central, the equality follows from Theorem 4. On the other hand, if is a field and is the ring of all the eventually constant sequences of elements of , then the set of all the sequences of elements of with finite support, , is an idempotent ideal of which is pure and satisfies that . Moreover, one has and then is right splitting. Then, by the Theorem 5.2 the TTF-triple in associated to is right splitting, but it is not centrally split.

References

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Pedro Nicolás
Departamento de Matemáticas,
Universidad de Murcia,
Aptdo. 4021,
30100 Espinardo, Murcia, España
pedronz@um.es

Manuel Saorín
Departamento de Matemáticas,
Universidad de Murcia,
Aptdo. 4021,
30100 Espinardo, Murcia, España
msaorinc@um.es

Recibido: 31 de enero de 2007
Aceptado: 1 de abril de 2007