Pure Injective Modules over Commutative Valuation Domains

Puninski, Gena

doi:10.1007/978-94-010-0652-1_13

Cited by 3 publications

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“…However, it is quite possible that a superdecomposable pureinjective module has type I . For instance, by [18], this is the case for every pure-injective module over a commutative valuation domain. Thus, type I is the easiest instance of superdecomposability.…”

Section: Discussionmentioning

confidence: 97%

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

Puninski¹

2008

Journal of Pure and Applied Algebra

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We construct an element in a direct product of finite dimensional modules over a string algebra such that the pure-injective envelope of this element is a superdecomposable module.A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For example, let R be the endomorphism ring of a countable dimensional vector space V , and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ∼ = e R ⊕ (1 − e)R as a right module over itself and R ∼ = e R ∼ = (1 − e )R . Furthermore, every nontrivial decomposition of R as a right module is of this form; therefore R is superdecomposable.For more examples, let R = k X, Y be a free algebra over a field k, and let E = E(R R ) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that R R has no nonzero uniform submodules. Therefore the same is true for E, and hence E is a superdecomposable injective module.More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pure-injective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pure-injective if M is a direct summand of a direct product of finite dimensional A-modules.If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pure-injective A-module has an indecomposable direct summand. But recently, Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pure-injective module (note that every string algebra is tame). The main drawback of his proof is that it depends on the cardinality of the ground field and is highly nonconstructive. Specifically, it is based on a quite ingenious construction of Ziegler [26] which seems to work only for countable fields.

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Section: Discussionmentioning

confidence: 97%

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

Puninski¹

2008

Journal of Pure and Applied Algebra

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“…Puninski, see e.g. [47], also has developed some ways of understanding the structure of these modules and has formulated conjectures about their structure. There seems, however, to have been rather limited development in this direction.…”

Section: Final Remarksmentioning

confidence: 99%

Superdecomposable pure-injective modules

Prest

2014

Advances in Representation Theory of Algebras

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“…In [134] information about the continuous pure-injective modules over a commutative valuation domain is obtained.…”

Section: Serial Ringsmentioning

confidence: 99%

Model theory and modules

Prest

2003

Handbook of Algebra

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The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se.Our default is that the term "module" will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the notation R-Mod denotes the category of left R-modules. By Ab we mean the category of abelian groups.In Part 1 we introduce the general concepts and in Part 2 we discuss these in more specific contexts.References within the text, as well as those in the bibliography, are neither complete nor comprehensive but are intended to lead the reader to a variety of sources. PurityPurity (pure embeddings, pure-injective modules) undoubtedly plays the central role so we will start with that. The notion of a pure embedding between modules was introduced by Cohn [30]. We say that the module A is a pure submodule of the module B if every finite system i x i r ij = a j (j = 1, ..., m) of R-linear equations with constants from A (so r ij ∈ R and a j ∈ A) and with a solution in B has a solution in A (a solution being elements b 1 , ..., b n such that i b i r ij = a j for all j). We extend this in the obvious way to define the notion of a pure embedding between modules and we also say 1 that an exact sequence 0 −→ A f − → B −→ C −→ 0 is pure-exact if f is a pure embedding.Functor categories Let D(R) = (R-mod, Ab) denote the category of additive functors (from now on we use the term "functor" to mean additive functor) from the category of finitely presented left modules to the category of abelian groups. This is a Grothendieck abelian category. It has a generating set consisting of finitely presented objects: indeed, the representable functors (L, −) for L ∈ R-mod are the finitely generated projective objects and, together, are generating. This category is locally coherent -any finitely generated subfunctor of a finitely presented functor is itself finitely presented -and of global dimension less than or equal to 2. A functor is finitely presented iff it is the cokernel of a map between two representable functors. The full subcategory C(R) = (R-mod, Ab) fp of finitely presented functors is an abelian category and the inclusion of (R-mod, Ab)fp into (R-mod, Ab) preserves exact sequences. Notice that the category (R-mod, Ab) is just the category of "modules" over the "ring with many objects" R-mod (better, over a small version of this). Concepts for modules over a ring generally make good sense in this context and largely can be understood in this way (that is, as having the same content that they have for modules). There is a full embedding of Mod-R into (R-mod, Ab) which is given on objects by sending M ∈ Mod-R to the functor M ⊗ R − : R-mod −→ Ab and which is given in the natural way on morphisms. The image o...

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Pure Injective Modules over Commutative Valuation Domains

Cited by 3 publications

References 5 publications

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

Superdecomposable pure-injective modules

Model theory and modules

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