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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.3 Bahía Blanca 2007
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
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
for all
in
and
in
.
- If
for all
in
then
is in
.
- 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
.
3.1. 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:
- Idempotent two-sided ideals
of
.
- 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:
- (Ideals of
generated by single) central idempotents of
.
- 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:
![~ {TTF -triples} = {idempotent ideals} 𝔏 ≃? ℜ ≃? ~ ℭ = {central idempotents }](/img/revistas/ruma/v48n3/3a0582x.png)
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:
is a TTF-class if and only if
for some idempotent
of
.
is a TTF-class if and only if
for some idempotent
of
.
is closed under submodules if and only if
is a flat right
-module.
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
- Left split TTF-triples in
.
- 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
![[ C 0 ] [ (1 - e)A(1 - e) 0 ] := M B eA(1 - e) eAe](/img/revistas/ruma/v48n3/3a05130x.png)
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:
- For every bimodule structure
and every right
-module
, the right
-module
is hereditary projective.
- There exists a bimodule structure
such that for every right
-module
the right
-module
is hereditary projective.
- Put
. If
is the minimal injective cogenerator of
, then the right
-module
is hereditary
-projective.
- The character module
is a hereditary
-projective right
-module.
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:
- 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:
- Right split TTF-triples in
whose associated idempotent ideal
is finitely generated on the left.
- 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:
is semiperfect.
- 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
![[ ] [ ] C (1 - e)A(1 - e) 0 M B = (1 - e)Ae eAe .](/img/revistas/ruma/v48n3/3a05215x.png)
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:
- Right split TTF-triples in
.
- 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:
- it is pure as a left ideal,
,
- it satisfies one of the following two equivalent conditions:
- for every integer
and every
-saturated right ideal
of
, there exists
such that
,
- for every integer
and every
-saturated submodule
of
, the quotient
is projective as a right
-module.
- for every integer
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:
- Right split TTF-triples in
.
- 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
![~ [ C 0 ] I → M 0](/img/revistas/ruma/v48n3/3a05287x.png)
of
![~ [ C 0 ] A → M H](/img/revistas/ruma/v48n3/3a05288x.png)
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.
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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