Pure Semisimplicity Conjecture Research Articles

Pure semisimplicity conjecture and Artin problem for dimension sequences

Inspired by a recent paper due to José Luis García, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight , of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding F ↪ M n ( G ) exists provided that n < 5 . As a byproduct, we obtain a division ring extension G ⊆ F such that the bimodule F F G has the right dimension sequence ( 1 , 2 , 2 , 2 , 1 , 4 ) . Finally, we formulate Conjecture A , which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture.

Small potential counterexamples to the pure semisimplicity conjecture

The pure semisimplicity conjecture or pssc states that every left pure semisimple ring has finite representation type. Let [Formula: see text] be division rings, and assume we identify conditions on a [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] which are sufficient to make the triangular matrix ring [Formula: see text] into a left pure semisimple ring which is not of finite representation type. It is then said that those conditions yield a potential counterexample to the pssc. Simson [17–20] gave several such conditions in terms of the sequence of the left dimensions of the left dual bimodules of [Formula: see text]. In this paper, conditions with the same purpose are given in terms of the continued fraction attached to [Formula: see text], and also through arithmetical properties of a division ring extension [Formula: see text].

Infinite dimensional serial algebras and their representations

We study arbitrary algebras all of whose indecomposable finite dimensional representations are uniserial, or more generally, comodules over serial coalgebras. We show that the infinite representation theory for such algebras parallels and non-trivially extends the theory of primary infinite abelian groups: we obtain a Kulikov and a Prüfer decomposability criterion and determine indecomposables in the category of locally finite modules over such an algebra. As applications, we obtain consequences on the infinite representation theory of the extended affine Dynkin quiver An˜, and the infinite line and half line quivers, and obtain Jordan form results for operators on infinite dimensional spaces. We also find a class of counterexamples to an infinite version of the pure semisimplicity conjecture: there exist algebras all of whose locally finite left modules are direct sums of indecomposables, but not all of whose right locally finite modules decompose as such. On the way, we also provide classifications of one-sided serial (co)algebras, and introduce and classify a class of quantum groups whose indecomposable representations are uniserial, namely, coserial pointed Hopf algebras. These are one-parameter deformations of group algebras and are obtained as certain Ore extensions of these.

On hereditary artinian rings and the pure semisimplicity conjecture: rigid tilting modules and a weak conjecture

A weak form of the pure semisimplicity conjecture is introduced and characterized through properties of matrices over division rings. The step from this weak conjecture to the full pure semisimplicity conjecture would be covered by proving that there do n

Mittat-Leffler conditions on modules

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.

The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

ABSTRACT Let G be a division ring and F ⊂ G a fixed subdivision ring of G such that dim F G = ∞, dim G F = 2 and the F-G-bimodule F G G has the infinite dimension type d−∞( F G G ) = (…, 2, 2…,2, 2, 1, ∞) in the sense of Simson (19962000), see Section 2. In particular, the hereditary ring is a counter-example to the pure semisimplicity conjecture. Let be a complete set of representatives of pairwise non-isomorphic preinjective right R G -modules. We prove that the reduced product is a direct sum , where , , s 0 = s 1 = (card G)ℵ0 , and means the direct sum of s j copies of the right ideal P j of R G . We also show that is isomorphic to ∏ℵ0 P 1. Communicated by J. Gomaz Pardo.

Endofinite modules and pure semisimple rings

Let R be a right pure semisimple ring, i.e., a ring R such that every right R-module is a direct sum of finitely generated modules. It is proved that R is of finite representation type if and only if every finitely presented (indecomposable) right R-module is endofinite, if and only if every finitely presented right R-module has a left artinian endomorphism ring. As applications, we obtain an alternative proof of the pure semisimplicity conjecture for PI-rings, and new criteria for a right pure semisimple ring to be of finite representation type.

On local right pure semisimple rings of length two or three

We investigate in this paper non-commutative local rings of the smallest length that are potential counter-examples to the pure semisimplicity conjecture. Throughout the paper is an associative ring with an identity element. We call local, if the Jacobson radical ( ) of is a two-sided maximal ideal. We denote by mod( ) the category of finitely generated right -modules. Given a right -module of finite length we denote by ( ) the length of . We recall that a ring is said to be of finite representation type, if is artinian and the number of the isomorphism classes of finitely generated indecomposable right (and left) -modules is finite. Following [24] we call a ring right pure semisimple, if every right -module is a direct sum of finitely generated modules. It is well known that a ring is of finite representation type if and only if is right pure semisimple and is left pure semisimple (see [2], [11], [18], [22]–[24]). It is still an open question, called the pure semisimplicity conjecture, if a right pure semisimple ring is of finite representation type (see [2] and [24], [25], [28]). In [13] the question is answered in affirmative for rings satisfying a polynomial identity and for self-injective rings (see also [7], [19] and [31]). The reader is referred to [42] and to the author’s expository papers [30] and [32] for a basic background and historical comments on the pure semisimplicity conjecture. It was shown by the author in [28] and [33] that there is a chance to find a counter-example to the pure semisimplicity conjecture and might be hereditary with two simple non-isomorphic modules. The existence of a counter-example depends on a generalized Artin problem on division ring extensions. In the present paper we are mainly interested in the existence of counter-examples to pure semisimplicity conjecture that are local of the smallest length, that is, of length ( ) two or three. This continues our study started in [28], [35] and [33]. It is shown in Lemma 3.1 that every such a local ring has ( )2 = 0. Therefore we study representation-infinite right pure semisimple local rings with ( )2 = 0 such that the Auslander-Reiten quiver (mod ) is of the form · · · → •→ •→ •→

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT In his recent work, [1] and [2], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.

Direct products of modules and the pure semisimplicity conjecture. Part II

We prove that the module categories of Noether algebras (i.e., alge- bras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product Q n2N Mn of finitely generated indecomposable modules Mn is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the Mn. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

ON SMALL RIGHT PURE SEMISIMPLE RINGS AND THE STRUCTURE OF THEIR AUSLANDER-REITEN QUIVER

Let R be an associative ring with an identity element. One of the main aims of this paper is to show how potential counter-examples R to the pure semisimplicity conjecture of length ℓ(RR ) two or three can look like, and the shape of their Auslander-Reiten quiver is described. *Partially supported by Polish KBN Grant 2 P0 3A 012 16

ON SMALL RIGHT PURE SEMISIMPLE RINGS AND THE STRUCTURE OF THEIR AUSLANDER-REITEN QUIVER

Let R be an associative ring with an identity element. One of the main aims of this paper is to show how potential counter-examples R to the pure semisimplicity conjecture of length ℓ(RR ) two or three can look like, and the shape of their Auslander-Reiten quiver is described.

An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II

An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II

An Artin problem for division ring extensions and the pure semisimplicity conjecture I

An Artin problem for division ring extensions and the pure semisimplicity conjecture I

On right pure semisimple hereditary rings and an Artin problem

The pure semisimplicity conjecture (pss R ) stated below is studied in the paper mainly for hereditary rings R. One of our main results is Theorem 3.6 containing various conditions which are equivalent to the conjecture (pss R ) for hereditary rings R. It follows from our main results together with recent results of Herzog [16] that in order to prove (pss R ) for any R it is sufficient (and necessary) to construct an indecomposable module of infinite length over any hereditary ring R of the form ( 0 G F FMF G ), where F, G are division rings and F M G is a simple F- G-bimodule such that dim M G is finite and dim f M is infinite (see Corollary 5.1). Moreover, the existence of a counterexample R to the pure semisimplicity conjecture is equivalent to a generalized Artin problem for division rings (see 4.3–4.6), which is much more difficult than the Artin problem for division ring extensions solved by Cohn in [5] and by Schofield in [20]. It may frighten people of finding an easy solution to the pure semisimplicity problem. On the other hand, it is concluded in Section 5 that studying generalized Artin problems can help solve the pure semisimplicity conjecture.