Left Pure Semisimple Ring Research Articles

Superstability, noetherian rings and pure-semisimple rings

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings.We characterize noetherian rings via superstability of the class of left modules with embeddings. Theorem 0.1For a ring R the following are equivalent.(1)R is left noetherian.(2)The class of left R-modules with embeddings is superstable.(3)For everyλ≥|R|+ℵ0, there isχ≥λsuch that the class of left R-modules with embeddings has uniqueness of limit models of cardinality χ.(4)Every limit model in the class of left R-modules with embeddings is Σ-injective.We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings. Theorem 0.2For a ring R the following are equivalent.(1)R is left pure-semisimple.(2)The class of left R-modules with pure embeddings is superstable.(3)There existsλ≥(|R|+ℵ0)+such that the class of left R-modules with pure embeddings has uniqueness of limit models of cardinality λ.(4)Every limit model in the class of left R-modules with pure embeddings is Σ-pure-injective.Both equivalences provide evidence that the notion of superstability could shed light in the understanding of algebraic concepts.As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

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].

FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

Splitting torsion pairs over pure semisimple rings

Let R be a left pure semisimple ring, and (D,C) be a splitting torsion pair of R-ind, i.e. (D,C) is a partition of the family of all indecomposable left R-modules such that HomR(D,C)=0 whenever D∈D and C∈C. Suppose further that D contains all indecomposable injective left R-modules. We show that for each module M in C, the endomorphism ring of M is a division ring and ExtR1(M,M)=0. Let W be the direct sum of all Ext-injective modules in C and all indecomposable projective modules in D. If W is endofinite, then there is an Ext-injective module in C which is the source of a left almost split morphism in R-mod. It is also proved that W is a tilting module, and if R is hereditary, then W has a hereditary endomorphism ring. As consequences, we recover with new proofs several recent results on left pure semisimple rings R. When R is left pure semisimple hereditary indecomposable, splitting torsion pairs and tilting modules over R can be characterized using the Ext-injective partition of R-mod. In particular, when R is left pure semisimple hereditary indecomposable with only two simple modules, we give a complete description of the distribution of indecomposable left R-modules, and their Gabriel–Roiter measures.

Auslander–Reiten components over pure-semisimple hereditary rings

Let R be a hereditary, indecomposable, left pure-semisimple ring. We show that R has finite representation type if and only if a certain finitely presented module is endofinite, namely, the tilting and cotilting module W studied in L. Angeleri Hügel (2007) [2]. We then apply the tilting and the cotilting functors to study the endomorphism ring of W and its Auslander–Reiten components. Finally, we transfer this information to the category of right R-modules.

On the endofiniteness of a key module over pure semisimple rings

Let R R be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R R -mod, and let W W be the left key R R -module; i.e., W W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R R -modules. We show that if the module W W is endofinite, then R R is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.

PURE SEMISIMPLE FINITELY ACCESSIBLE CATEGORIES AND HERZOG'S CRITERION

Let [Formula: see text] be a finitely accessible category with products, and assume that its symmetric category [Formula: see text] is also finitely accessible and pure semisimple. We study necessary and sufficient conditions in both categories for [Formula: see text] (and hence [Formula: see text]) to be of locally finite representation type. In particular, we obtain a generalization of Herzog's criterion for finite representation type of left pure semisimple and right artinian rings. As an application, we prove that a left pure semisimple ring R with enough idempotents which has a self-duality is of locally finite representation type if and only if it is left locally finite.

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R , it is shown that the local duality establishes a bijection between the preinjective left R -modules and the preprojective right R -modules, and any preinjective left R -module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R -mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R -modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361–376] for hereditary rings.

A key module over pure-semisimple hereditary rings

Let R be a hereditary, indecomposable, left pure semisimple ring. Inspired by [I. Reiten, C.M. Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006) 180–224], we investigate the perfect cotorsion pair ( C , D ) in R - Mod generated by the preinjective component q. We show that there is a finitely generated product-complete tilting and cotilting left R-module W such that C = Cogen W and D = Gen W . The finite subcategory w of R - mod given by the indecomposable summands of W stores important information on R. For example, if we assume R of infinite representation type, then by [B. Zimmermann-Huisgen, Strong preinjective partitions and representation type of artinian rings, Proc. Amer. Math. Soc. 109 (1990) 309–322] there are non-preinjective indecomposable modules occurring as direct summands of products of preinjective modules, and it turns out that w is precisely the class of such modules. Moreover, we prove that R has finite representation type if and only if every module in w is source of a left almost split map in R - mod . Finally, we address the question when W is endofinite.

Strong preinjective partitions and almost split morphisms

A family C of finitely generated indecomposable modules is said to have right almost split morphisms if there is a right almost split morphism f : N→M in add C for each indecomposable module M in C . We show that, for any ring R, if C is a family of indecomposable left R-modules of finite length such that C contains a finite cogenerating set and every subfamily of C has right almost split morphisms, then every subfamily of C has a unique strong preinjective partition of countable length. Sufficient conditions are given for a family of Noetherian modules with local endomorphism rings to have the property that each of its subfamilies has right almost split morphisms. A consequence of our results is a theorem, obtained by Zimmermann-Huisgen (Proc. Amer. Math. Soc. 109 (1990) 309–322) with different techniques, which asserts that every family of finitely generated indecomposable left modules over a left pure semisimple ring has a unique strong preinjective partition of countable length.

Preinjective modules and finite representation type of artinian rings

Let R be a left artinian ring such that every finitely presented right .ft-module is of finite endolength. It is shown that the cardinality of the set of isomorphism classes of preinjective right R-modules is less than or equal to the cardinality of the set of isomorphism classes of preprojective left R-modules, and R is of finite representation type if and only if these cardinal numbers are finite and equal to each other. As a consequence, we deduce a theorem, due to Herzog [17], asserting that a left pure semisimple ring R is of finite representation type if and only if the number of non-isomorphic preinjective right R-modules is the same as the number of non-isomorphic preprojective left .R-modules. Further applications are also given to provide new criteria for artinian rings with self-duality and artinian Pi-rings to be of finite representation type, which imply in particular the validity of the pure semisimple conjecture for these classes of rings.

A test for finite representation type

Let R be a left pure-semisimple ring. It is proved that if R has self-duality or if R is a polynomial identity ring, then R is of finite representation type. If there exists an example of a left pure-semisimple ring which is not of finite representation type, we show that then there exists an example which is hereditary, but not right artinian. We reduce the question of the existence of such an example to a problem regarding simple bimodules over division rings.

Finitely Presented Right Modules Over a Left Pure-Semisimple Ring

j ^)- Zimmermann-Huisgen [22, Corollary 2]showed that these are the left pure-semisimple rings defined below. And although leftpure-semisimple rings have many attractive features (see [15, p. 210]), it is not knownwhether they are all of finite representation type, that is, left artinian with only finitelymany indecomposable left modules.Auslander [1] showed that every left pure-semisimple artin algebra is of finiterepresentation type (an artin algebra is a ring which is finitely generated as a moduleover an artinian centre). If A is an artin algebra an

On modules with the Kulikov property and pure semisimple modules and rings

For an R-module M let σ[ M] denote the category of submodules of M-generated modules. M has the Kulikov property if submodules of pure projective modules in σ[ M] are pure projective. The following is proved: Assume M is a locally noetherian module with the Kulikov property and there are only finitely many simple modules in σ[ M]. Then, for every n ϵ N , there are only finitely many indecomposable modules of length ⩽ n in σ[ M]. With our techniques we provide simple proofs for some results on left pure semisimple rings obtained by Prest and Zimmermann-Huisgen and Zimmermann with different methods.