# Finite Rank Torsion Free Modules Over Dedekind Domains

In Math Reviews, abelian group theory sits ensconced among all the other varieties of group theory, having no more in common with its neighbors than a handful of words such as "group," "subgroup," and "homomorphism."

Kaplansky, in his "little red book", asserted that abelian group theory is really the study of modules over principal ideal domains, and since then most abelian group theorists tend to feel more at home with commutative ring theory than with group theory in general.

On the other hand, if abelian group theory is a branch of commutative ring theory, it is certainly a highly specialized branch. In the study of torsion groups, the ring theorist finds more concepts that are strange to him than concepts which are familiar, and this cannot be remedied by a mere change in terminology.

The downside of the influence of Kaplanksy's statement was that it promoted the idea that it was unnecessary and perhaps even pretentious to state results for modules over principal ideal domains, since results for abelian groups would automatically carry over. To some extent this attitude was justified as long as abelian group theory was largely the study of torsion modules, although even in Kaplansky's book it was not totally clear how to translate the proof of Ulm's Theorem, using Kaplanksy's back-and-forth method, into the context of modules over an uncountable ring.

In the realm of torsion free modules, however, the idea that results for abelian groups will automatically hold for modules over arbitrary principal ideal domains is simply not tenable. The theory of finite rank torsion free abelian groups is full of results that depend on countability, or on having characteristic zero, or working over a ring whose quotient field is a perfect field, as well as proofs using quite specialized results from number theory. Among these is your author's theorem that for a finite rank torsion free abelian group G there are, up to isomorphism, only finitely many groups isomorphic to summands of G (see Chapter 11). This is not true for modules over an arbitrary principal ideal domain.

The transition from modules over principal ideal domains to modules over dedekind domains is actually a much smaller leap than that from the integers to an arbitrary PID. Moreover in the process of reformulating theorems and proofs to be valid over dedekind domains, one sees these results in a new and -- in your author's opinion -- "more correct" way.

And one becomes more aware of the fact that the theory of finite rank torsion free abelian groups is moving away from abelian group theory in general in much the same fashion that abelian group theory has moved away from general group theory. Unlike the theory of torsion groups, the theory of finite rank torsion free modules is becoming something that fits in fairly well with the mainstream of commutative ring theory.

The only thing that has prevented this from being apparent has been terminology. In this book, I have used for the most part terminology familiar to commutative ring theorists. Abelian groups theorists will have to make some minor adjustments. Eventually I want to supply a brief table to facilitate such adjustments. (For instance, the word "finite" in abelian group theory should be replaced with "finite length" for module theory.)

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Many of these chapters are essentially complete, but they all need at least a little remedial work.
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A lot of the files listed below are in
PDF (Adobe Acrobat) format. Alternate versions are in DVI format (produced by TeX; see see here for a DVI viewer provided by John P. Costella) and postscript format (viewable with ghostscript.) Some systems may have some problem with certain of the documents in dvi format, because they use a few German letters from a font that may not be available on some systems. (Three alternate sites for DVI viewers, via FTP, are CTAN, Duke, and Dante, in Germany.)
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## 0. Modules Over Commutative Rings (31 pages)

( Cick here for DVI version.)

Localization and Globalization
Comaximal Ideals
Associated Primes
Modules with Finite Length
Rank-One Projective Modules
Dedekind Domains
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## 1. Fundamentals (24 pages)

Cheap Definitions
Divisible and p-Divisible Modules
Purity
Essential Submodules
Finitely Generated Modules
Rank and p-Rank
Finite Rank Torsion Free Rings
Tensor Products and Hom
Indecomposable Modules
Complete Modules
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## 2. Rank-one Modules and Types (24 pages)

Rank-one Modules
Types
Idempotent Types
The Type of an Element
The Typeset and Cotypeset
Characteristics
Richman Type
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## 3. Quasi-Homomorphisms (23 pages)

The Category of Torsion Free Modules under Quasi-homomorphisms
Quasi-concepts
Quasi-direct Decompositions
Jonsson's Theorem
More Stolen Theorems
Murley Modules
Murley Rings
The Torsion Submodule of Ext
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## 4. The t-Socle and t-Radical (19 pages)

Domination
t-Saturated and t-Bounded Modules
Baer's Lemma
Completely Decomposable Modules
Warfield Isomorphism Theorems
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## 5. Butler Modules (33 pages)

Balanced and Co-balanced Submodules
Proof of Butler's Theorem
Typeset and Cotypeset
Critical Types
Cocritical Types
Hom and Tensor Product
Almost Completely Decomposable Modules
Regulating Submodules
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## 6. Splitting Rings and Splitting Fields (29 pages)

The Pontryagin Module
The Kurosh Theorem
A Reverse Perspective
Splitting Fields
Corner's Theorem
Splitting Rings
Butler Modules
Extension of Scalars
Arnold Duality (Local Case)
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## 7. Torsion Free Rings (33 pages)

General Results
Prime & Semi-prime Rings
Quasi-products of Rings
Field of Definition
Strongly Indecomposable Domains
Composita
Valuation Rings
Quasi-separable Integral Domains Are R-split
When Is a Module Isomorphic to an Integral Domain?
Strongly Homogeneous Modules
Multiplications on a Strongly Indecomposable Module
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## 8. Quotient Divisible Modules (12 pages)

Properties of Quotient Divisible Modules
The Local-Global Principle
The Quotient Divisible Core
Quotient Divisible Murley Modules
Arnold Duality (Global Case)
Generalized Butler Modules
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## 9. Locally Free Modules (13 pages)

Locally Free Modules & Quotient Divisible Modules
Warfield Duality
Modules With Non-trivial Trace
Quasi-summands of Tensor Products
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## 11. Direct Sum Decompositions (45 pages)

Fully Invariant Summands
The Exchange Property
C-Projective Modules
The Baer Splitting Property
The Substitution Property
Stable Range
Finite Rank Torsion Free Modules
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## Bibliography on torsion free abelian groups

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[ A Graduate Algebra Course | Topics in Homological Algebra ]

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