On finitely generated projective modules and multiplication modules

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in January 1987.     

关于有限生成投射模为自由模的环

若环R上一切有限生成投射模都是自由模,则称R为PF环,本文讨论了PF环的一些性质及其一些应用。     

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

Endomorphism rings and the category of finitely generated projective modules

Given a left R-module M, we study the connection between the (right and left) properties of its endomorphism ring S=End(_RM) and the properties of the category σ _f [M] of all submodules of finitely M-generated left R-modules.     

On finitely generated modules over quasi-Euclidean rings

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let \(\text {E}_n(R)\) be the subgroup of \(\text {GL}_n(R)\) generated by the elementary matrices. In this paper we study the action of \(\text {E}_n(R)\) by matrix multiplication on the set \(\text {Um}_n(M)\) of unimodular rows of M of length \(n \ge k\). Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if \(n > k\). We also prove that \(\text {Um}_k(M) /\text {E}_k(R)\) is equipotent with the unit group of \(R/\mathfrak {a}_1\) where \(\mathfrak {a}_1\) is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.     

Fuzzy finitely generated modules

We introduce the category of fuzzy modules and discuss the construction of fuzzy finitely generated modules. Characterizations of a sort of fuzzy modules, in which the fuzzy value distribution is finite, are partly obtained.     

Decomposing finitely generated torsion modules

Let R be a commutative ring and let F be a filter of ideals of R. We give three characterizations of the rings with the property that all finitely generated F pretorsion R modules decompose into a direct sum of cyclic R submodules. One of these characterizations involves decomposing the filter F into a product of subfilters. This filter decomposition is shown to be unique, and is used to characterize several other ring properties relative to the filter of ideals.     

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.

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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.

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Lavoro eseguito nell'ambito del GNSAGA.     

Reduced rings whose finitely generated modules decompose

This paper proves Hua's Theorem and, from this, the Fundamental Theorem of Projective Geometry for the projective line over a commutative ring with “many units”.     

Quasi-isomorphisms of finitely generated modules over valuation domains

Sunto Sia R un anello commutativo. Due R-moduli si dicono quasi-isomorfise ciascuno è isomorfo ad un sottomodulo dell'altro. Si prova che, a meno di quasi-isomorfismi, un R-modulo si decompone in modo unico in una somma diretta finita di sottomoduli uniformi. Nel caso in cui R è un dominio di valutazione, si prova che, se X e Y appartengono ad una certa classe di R-moduli finitamente generati e uniformi, X e Y sono quasiisomorfi se e solo se sono isomorfi. Usando i precedenti risultati, otteniamo nuove classi di R-moduli finitamente generati che, a meno di isomorfismi, si decompongono in modo unico in somma diretta di addendi indecomponibili.

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Lavoro eseguito con il contributo del Ministero della Pubblica Istruzione.     

On projective modules

LetR be a commutative Noetherian ring with identity. The Hermite dimension ofR is defined to be the least integerr such that every stably freeR-module of rank greater thanr is free. In this paper we study ringsR obtained upon inversion of elements of a given ringA. We show that the Hermite dimension ofR does not depend on the Hermite dimension ofA, it depends on the Krull dimension ofA.     

Finitely generated projective modules over exchange rings

This paper studies finitely generated projective modules over exchange rings. We prove that cancellation holds inp(R), andK _o (R) is completely determined by the continuous maps from the spectrum ofR toZ ifR is an exchange ring andR/J(R) is a ring with central idempotent elements.     

Flat modules and rings finitely generated as modules over their center

A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:

1. A is a right or left distributive semiprime ring;
2. for any maximal idealM of a subringR central inA, the ring of quotientsA _M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
3. all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.