Baer modules over valuation domains

Summary A module B over a commutative domain R is said to be a Baer module if Ext _R ¹ (B, T)=0for all torsion R-modules T. The case in which R is an arbitrary valuation domain is investigated, and it is shown that in this case Baer modules are necessarily free. The method employed is totally different from Griffith's method for R=Z which breaks down for non-hereditary rings.This research was partially supported by NSF Grants DMS-8400451 and DMS-8500933.     

On divisible modules over domains

This paper is part of the author's Ph.D. dissertation written under the direction of Professor L. Fuchs at Tulane University.     

A note on linear transformations over integral domains

It is an easy fact from linear algebra that if M is a finite-dimensional vector space over a field R, ϕM→M a diagonalizable linear transformation, and N a ϕ-invariant subspace of M, then ϕ∣N is diagonalizable. We show that an appropriate generalization of this holds for M a torsion-free module over an integral domain R.     

On two-generated modules over valuation domains

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Exchanging torsion modules over Dedekind domains

The author was partially supported by a grant (Project 1419) from the Research Grants Committee of the University of Alabama.     

On weak-injective modules over integral domains

We show that a weak-injective module over an integral domain need not be pure-injective (Theorem 2.3). Equivalently, a torsion-free Enochs-cotorsion module over an integral domain is not necessarily pure-injective (Corollary 2.4). This solves a well-known open problem in the negative.In addition, we establish a close relation between flat covers and weak-injective envelopes of a module (Theorem 3.1). This yields a method of constructing weak-injective envelopes from flat covers (and vice versa). Similar relation exists between the Enochs-cotorsion envelopes and the weak dimension ?1 covers of modules (Theorem 3.2).     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).