3 edition of **Torsion modules over Dedekind prime rings** found in the catalog.

Torsion modules over Dedekind prime rings

David Eisenbud

- 181 Want to read
- 26 Currently reading

Published
**1970**
.

Written in English

Classifications | |
---|---|

LC Classifications | Microfilm 29540 |

The Physical Object | |

Format | Microform |

Pagination | ii, 63 l. |

Number of Pages | 63 |

ID Numbers | |

Open Library | OL1250016M |

LC Control Number | 94895601 |

Integral closure of rings 13 How integral closure arises 14 Dedekind–Mertens formula 17 Exercises 20 2. Integral closure of rings 23 Basic facts 23 Lying-Over, Incomparability, Going-Up, Going-Down 30 Integral closure and grading 33 Rings of homomorphisms of ideals 39 Exercises 42 3. Separability. Also in , Eisenbud received his Ph.D. from Chicago for his thesis on non-commutative ring theory Torsion Modules over Dedekind Prime Rings. Eisenbud's first paper was not on ring theory, however, but rather on group theory with Groups of order automorphisms of certain homogeneous ordered sets ().

Classifying E-algebras over Dedekind domains cotorsion-free modules’ when the underlying ring R is a Dedekind domain. the P-primary component of the torsion submodule tP(A)=0 for each prime P ∈P, then A is an extension of an ideal I by a P-pure. Rank is defined over domains (or, more complicatedly, as an n-tuple over for generically free modules over arbitrary rings) by relating it to vector-space dimension of a localization. $\endgroup$ – Graham Leuschke Jun 30 '10 at

RINGS WHICH ADMIT FAITHFUL TORSION MODULES f n → be a continuous function with zero set −nn. Now deﬁne g n = f 1 f 2 f n, and let =g n n ∈ + (thus is the collection of the principal ideals generated by the g n). It is easily checked that n>0 R/g n is a faithful torsion. TORSION-FREE MODULES 3 Lemma (1) If X is inﬂnite then R~ is not a ﬂnitely generated R-module. (2) If R = Z[E1] then C = 0 and () fails. Proof. (1) Suppose instead that R~ is generated by f1; ;fn over a disjoint clopen cover fYj: 1 • j • tg of X such that each fijYj is constant, say kij 2 Z. Since X is inﬂnite, some Yj, say Y1, is distinct ﬁ;ﬂ 2 Y1.

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In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime can be shown that such a factorization is then necessarily unique up to the order of the factors.

There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition. 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.

Flat vs. torsion-free modules. Any flat module is torsion-free. The converse holds over the integers, and more generally over principal ideal domains. This Torsion modules over Dedekind prime rings book from the above characterization of flatness in terms of ideals.

Yet more generally, this converse holds over Dedekind rings. An integral domain is called a Prüfer domain if every. Browse other questions tagged commutative-algebra modules finitely-generated dedekind-domain or ask your own question.

The Overflow Blog The Overflow # Jokes on us. Torsion free modules In this section we discuss torsion free modules and the relationship with flatness (especially over dimension 1 rings).

Definition Modules over bounded Dedekind prime rings denotes the singular submo dule of M A (that is, the set of al l elements m ∈ M su ch that r (m) is an essential right id eal of the ring A). Buy Foundations of Commutative Rings and Their Modules (Algebra and Applications (22)) Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory.

Chapter 7 deals with multiplicative ideal theory over integral by: 6. This book discusses as well the non-finitely generated cases and studies infinitely generated projective modules.

The final chapter deals with abelian groups G that are injective when viewed as modules over their endomorphism rings E(G).

This book is a valuable resource for mathematicians. Browse other questions tagged abstract-algebra modules ideals localization dedekind-domain or ask your own question.

Featured on Meta Improving the Review Queues - Project overview. $\begingroup$ Just to make your answer complete: Lam proves in Theorem (see also Prop. ) in his "Lectures on Modules and Rings" (Springer GTM ) that a (commutative) domain is a Prüfer ring if and only if every torsion-free module is flat.

$\endgroup$ – Theo Buehler Jan 4 '11 at For modules over rings. Given a ring R R, an element m m in an R R-module M M is torsion element if there is a nonzero element r r in R R such that r m = 0 r m=0.

A torsion module is a module whose elements are all torsion. A torsion-free module is a module. TORSION FREE RINGS E. Lady J In Chapter 1 we considered the problem of constructing, for a given prime p,a p-reduced module Gwith rankG>1 and p-rankG= 1. Examplethe Pontryagin module, was an example of such a construction, and it ultimately led us to the concept of splitting elds and splitting rings.

Another approach is. To be more precise, the 1-tilting torsion classes over a Dedekind domain R coincide with the classes of -divisible modules, where denotes an arbitrary set of maximal ideals of R; in this setting a.

Lady, Grothendieck rings for certain categories of quasihomomorphisms of torsion free modules over dedekind domains, J. Algebra Google Scholar [12] E. Lady, Relations between Hom, Ext, and tensor product for certain categories of modules over dedekind domains, in Abelian Group Theory, Lecture Notes in Mathematics (), 53–Cited by: Abstract.

We obtain necessary and sufficient conditions in order that an arbitrary pure monoendomorphism of a module decomposed into a direct sum of rank 1 torsion-free modules over a Dedekind ring be an by: 1.

Torsion theories over commutative rings Paperback – January 1, by Willy Brandal (Author)Cited by: 3. The subject of torsion-free modules over an arbitrary integral domain arises naturally as a generalization of torsion-free abelian groups.

In this volume, Eben Matlis brings together his research on torsion-free modules that has appeared in a number of mathematical journals. Professor Matlis has reworked many of the proofs so that only an elementary knowledge of homological algebra and.

If R is a slender ring, then R[x] and M n (R) (the ring of n × n matrices over R) are slender rings. [Hint: use 2 and ] 5. A non-slender Dedekind domain is a p.i.d. [Hint: use the facts: If P is a prime ideal of a Dedekind domain R, then P is a maximal ideal and R P is a p.i.d.; also, a Dedekind domain with only finitely many primes is.

Revised August DAVID EISENBUD VITA Born April 8,New York City US Citizen Married, with two children EDUCATION B. University of Chicago M. University of Chicago Ph.

University of Chicago Advisors: Saunders MacLane, J. Robson Thesis: Torsion Modules over Dedekind Prime Rings POSITIONS HELD. Examples of torsion-free modules. Over a commutative ring R with total quotient ring K, a module M is torsion-free if and only if Tor 1 (K/R,M) ore flat modules, and in particular free and projective modules, are torsion-free, but the converse need not be true.

An example of a torsion-free module that is not flat is the ideal (x, y) of the polynomial ring k[x, y] over a field k. Finite Rank Torsion Free Modules Over Dedekind Domains - E. Lee Lady; University of Hawaii A book that discusses abelian group theory as not just another variety of group theory but the study of modules over principal ideal domains, a highly specialized branch of .TORSION FREE AND PROJECTIVE MODULES BY HYMAN BASS(1) Introduction.

Serre has asked [5, p. ] whether projective modules over the polynomial ring K[XI, * * *, X"], K a field, are free, and Seshadri has proved that this is so when n =2.

More precisely, he showed that every R[X]-projective module 'comes from" one over R if R is either a principal.Starting from definitions, the book introduces fundamental constructions of rings and modules, as direct sums or products, and by exact sequences.

It then explores the structure of modules over various types of ring: noncommutative polynomial rings, Artinian rings .