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Generic Modules Over Artin Algebras
 Proc. London Math. Soc
, 1995
"... this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a n ..."
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this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a new characterization of the pureinjective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pureinjective. Next we consider indecomposable endofinite modules. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. Changing slightly the original definition, we say that a module is generic if it is indecomposable endofinite but not finitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the nonfinitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PIrings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f : Mod(\Gamma) ! Mod() which commutes with direct limits and products, preserves certain finiteness conditions. For example, if a \Gammamodule M is endofinite then f(M) is endofinite. If in addition End \Gamma (M) is a PIring, then End (N) is a PIring for every indecomposable direct summand N of f(M ). This material is collected in Section 6 and 7. In Section 8 we introduce an effective method to construct generic modules over artin algebras from socalled generalized tubes. The special case of a tube in the AuslanderReiten quiver is discussed in t...
HOW TO CONSTRUCT A ‘CONCRETE’ SUPERDECOMPOSABLE PUREINJECTIVE MODULE OVER A STRING ALGEBRA
"... Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective 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 an examp ..."
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Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective 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 an 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 ′ ∼ = eR ′ ⊕ (1 − e)R ′ as a right module over itself and R ′ ∼ = eR ′ ∼ = (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(RR) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that RR has no nonzero uniform submodules. Therefore the same is true for E, 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 pureinjective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pureinjective if M is a direct summand of a direct product of finite dimensional Amodules. If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pureinjective Amodule has an indecomposable direct summand. But recently Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pureinjective module (note that every string algebra is tame). The main
TOWARDS THE DECIDABILITY OF THE THEORY OF MODULES OVER FINITE COMMUTATIVE RINGS
"... Abstract. On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with decidable theory of modules. We prove that if R is (finite length) wild, then the theory of all Rmodu ..."
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Abstract. On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with decidable theory of modules. We prove that if R is (finite length) wild, then the theory of all Rmodules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings. 1.
Hammocks for String Algebras
, 1998
"... this paper we consider the class of string algebras and deal with the corresponding problem. These algebras are usually representationinfinite and are regarded as an important class of tame algebras. We introduce a generalized notion of hammocks for string algebras and prove various combinatorial p ..."
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this paper we consider the class of string algebras and deal with the corresponding problem. These algebras are usually representationinfinite and are regarded as an important class of tame algebras. We introduce a generalized notion of hammocks for string algebras and prove various combinatorial properties of these posets. In case one is dealing with a representationfinite string algebra, our definition and the definition given in [RV] coincide. As a main application we compute the index of nilpotency (up to a small error term) of the radical of Amod where A is a string algebra. We also construct examples which show that every possible index occurs. Only very few examples of this type were known before.