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 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).     

On two-generated modules over valuation domains

Archiv der Mathematik -     

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.     

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).     

Indecomposable finitely generated modules over valuation domains

Summary LetR be a valuation domain,S a maximal immediate extension ofR. We introduce the definition of unitary independence. We use units ofS, which are unitarily independent over an ideal ofR, to construct indecomposable finitely generatedR-modules with Goldie dimension greater than one. We prove that, ifR is archimedean, the endomorphism ring of an indecomposable finitely generatedR-module is local. On the other hand, we prove that, ifR is a suitable non archimedean valuation domain, there exist indecomposable finitely generatedR-modulesM such that End (M) is not local.

---

Riassunto SiaR un dominio di valutazione,S un'estensione massimale immediate diR. Si introduce la definizione di indipendenza unitaria. Si usano unità diS unitariamente indipendenti su un ideale diR per costruireR-moduli finitamente generati indecomponibili con dimensione di Goldie maggiore di uno. Si dimostra che, seR è archimedeo, l'anello degli endomorfismi di unR-modulo finitamente generato indecomponibile è locale. Si prova altresì che, seR è un opportuno dominio di valutazione non archimedeo, esistonoR-moduliM finitamente generati indecomponibili, tali che End (M) non è locale.

---

---

Lavoro eseguito nell'ambito del GNSAGA.