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Some Algebraically Compact Modules. I
 I, Proceedings Padua Conference
, 1995
"... . Given a finite dimensional monomial algebra, one knows that some finite dimensional indecomposable modules may be described by words (finite sequences of letters) using as letters the arrows of the quiver and their formal inverses. To every word w, one can attach a socalled string module M(w). He ..."
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. Given a finite dimensional monomial algebra, one knows that some finite dimensional indecomposable modules may be described by words (finite sequences of letters) using as letters the arrows of the quiver and their formal inverses. To every word w, one can attach a socalled string module M(w). Here, we are going to construct certain infinite dimensional modules: We will consider Nwords and Zwords (thus infinite sequences of letters) satisfying suitable periodicity conditions. To every such Nword or Zword x, we describe an algebraically compact module C(x): This module C(x) is obtained from the corresponding string module M(x) as a kind of completion. Keywords. Algebraically compact modules, pure injective modules. Linearly compact modules. Quivers. Words, Nwords, Zwords. String modules, Pr ufer modules, padic modules. Special biserial algebras. Kronecker modules. Shift endomorphism. 1. Finite Words and Finite Dimensional Modules Let k be a field. Let A be a finite dimensio...
When a superdecomposable pureinjective module over a serial ring exists
"... Abstract. We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame ” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekindlike rings and their homomorphic images. We show that both are undefined in most cases. 1 ..."
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Abstract. We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame ” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekindlike rings and their homomorphic images. We show that both are undefined in most cases. 1
Vaught's Conjecture for Superstable Theories of Finite Rank
, 1993
"... In this paper we prove Vaught's conjecture for superstable theories in which each complete type has finite U \Gamma rank. The general idea is to associate with the theory an V \Gamma definable group G (called the structure group) which controls the isomorphism types of the models. ..."
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In this paper we prove Vaught's conjecture for superstable theories in which each complete type has finite U \Gamma rank. The general idea is to associate with the theory an V \Gamma definable group G (called the structure group) which controls the isomorphism types of the models.
DECIDABILITY OF THE THEORY OF MODULES OVER COMMUTATIVE VALUATION DOMAINS
"... Abstract. We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable. 1. ..."
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Abstract. We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all Vmodules is decidable. 1.
Pure injective envelopes of finite length modules over a Generalised Weyl Algebra
"... We investigate certain pureinjective modules over generalised Weyl algebras. We consider pureinjective hulls of finite length modules, the elementary duals of these, torsionfree pureinjective modules and the closure in the Ziegler spectrum of the category of finite length modules supported on a n ..."
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We investigate certain pureinjective modules over generalised Weyl algebras. We consider pureinjective hulls of finite length modules, the elementary duals of these, torsionfree pureinjective modules and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finitedimensional algebras and hereditary orders.
Krull–Gabriel dimension and the modeltheoretic complexity of the category of modules over group rings of finite groups
 J. London Math. Soc
"... Abstract. We classify group rings of finite groups over a field F according to the modeltheoretic complexity of the category of their modules. For instance we prove that, if F contains a primitive cubic root of 1, then the Krull–Gabriel dimension of such rings is 0, 2, or undefined. 1. ..."
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Abstract. We classify group rings of finite groups over a field F according to the modeltheoretic complexity of the category of their modules. For instance we prove that, if F contains a primitive cubic root of 1, then the Krull–Gabriel dimension of such rings is 0, 2, or undefined. 1.
Pureinjective Modules over Tubular Algebras and String Algebras, Doctoral Thesis
, 2011
"... We show that, for any tubular algebra, the lattice of ppdefinable subgroups of the direct sum of all indecomposable pureinjective modules of slope r has mdimension 2 if r is rational, and undefined breadth if r is irrational and hence that there are no superdecomposable pureinjectives of ration ..."
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We show that, for any tubular algebra, the lattice of ppdefinable subgroups of the direct sum of all indecomposable pureinjective modules of slope r has mdimension 2 if r is rational, and undefined breadth if r is irrational and hence that there are no superdecomposable pureinjectives of rational slope, but there are superdecomposable pureinjectives of irrational slope, if the underlying field is countable. We determine the pureinjective hull of every direct sum string module over a string algebra. If A is a domestic string algebra such that the width of the lattice of ppformulas has defined breadth, then classify “almost all ” of the pureinjective indecomposable Amodules. 8
The nonstandard quantum plane
"... Let k be a field of char(k) ̸ = 2, and suppose that q ∈ k is not a root of unity. The associated quantum plane [5, §IV.1], denoted by kq[x, y], is defined to be the free kalgebra k{x, y} generated by x and y, modulo the relation yx = qxy. The set of monomials {x i y j}i, j≥0 is a basis for the unde ..."
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Let k be a field of char(k) ̸ = 2, and suppose that q ∈ k is not a root of unity. The associated quantum plane [5, §IV.1], denoted by kq[x, y], is defined to be the free kalgebra k{x, y} generated by x and y, modulo the relation yx = qxy. The set of monomials {x i y j}i, j≥0 is a basis for the underlying kvector space, and for every pair (i, j) of nonnegative integers,
Towards decidability of the theory of pseudofinite dimensional representations of sl2(k
"... In this paper, we refine the analysis begun in Ivo Herzog’s paper [7] on representations of the Lie algebra sl2(k), where k is an algebraically closed field of characteristic 0. Our principal contribution is to bring out a connection to fundamental problems in the diophantine geometry of curves. We ..."
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In this paper, we refine the analysis begun in Ivo Herzog’s paper [7] on representations of the Lie algebra sl2(k), where k is an algebraically closed field of characteristic 0. Our principal contribution is to bring out a connection to fundamental problems in the diophantine geometry of curves. We expect to show, in a subsequent paper, that the theory