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.

In this paper we attempt to bridge a gap or, perhaps, to strengthen some existing links. Model theory has evolved in two sharply different directions. One is set-based, centred

We investigate certain pure-injective modules over generalised Weyl algebras. We consider pure-injective hulls of finite length modules, the elementary duals of these, torsionfree pure-injective 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 finite-dimensional algebras and hereditary orders.

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 Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases. 1

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 k-algebra 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 k-vector space, and for every pair (i, j) of nonnegative integers,

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

Abstract not found

Abstract. We study the first order theory of Bezout difference rings. In particular we show that rings of sequences very rarely have decidable theories as difference rings, or even decidable model completions.

Let M be a Verma module over the Lie algebra, sl2(k), of trace zero 2×2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U(sl2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules. 1

Abstract. We construct, by model-theoretic methods, several exponentiations on the universal enveloping algebra U of the Lie algebra sl2(C). MSC: 16S30, 17B10, 03C60. Key words: universal enveloping algebra, exponential map, asymptotic cone