Torsion-free modules by Eben Matlis

By Eben Matlis

The topic of torsion-free modules over an arbitrary crucial area arises certainly as a generalization of torsion-free abelian teams. during this quantity, Eben Matlis brings jointly his study on torsion-free modules that has seemed in a few mathematical journals. Professor Matlis has transformed a number of the proofs in order that merely an effortless wisdom of homological algebra and commutative ring conception is critical for an figuring out of the theory.

The first 8 chapters of the booklet are a normal advent to the speculation of torsion-free modules. This a part of the ebook is acceptable for a self-contained simple direction at the topic. extra really expert difficulties of discovering all integrally closed D-rings are tested within the final seven chapters, the place fabric lined within the first 8 chapters is applied.

An indispensable area is expounded to be a D-ring if each torsion-free module of finite rank decomposes right into a direct sum of modules of rank 1. After a lot research, Professor Matlis discovered that an integrally closed area is a D-ring if, and provided that, it's the intersection of at such a lot maximal valuation rings.

Show description

Read or Download Torsion-free modules PDF

Similar algebra books

Advanced Algebra: Along with a companion volume Basic Algebra

Uncomplicated Algebra and complicated Algebra systematically enhance ideas and instruments in algebra which are important to each mathematician, even if natural or utilized, aspiring or verified. jointly, the 2 books provide the reader a world view of algebra and its function in arithmetic as a complete. Key issues and contours of complex Algebra:*Topics construct upon the linear algebra, staff thought, factorization of beliefs, constitution of fields, Galois conception, and effortless conception of modules as constructed in simple Algebra*Chapters deal with a number of subject matters in commutative and noncommutative algebra, supplying introductions to the idea of associative algebras, homological algebra, algebraic quantity concept, and algebraic geometry*Sections in chapters relate the idea to the topic of Gröbner bases, the root for dealing with platforms of polynomial equations in laptop applications*Text emphasizes connections among algebra and different branches of arithmetic, quite topology and complicated analysis*Book includes on favorite topics habitual in uncomplicated Algebra: the analogy among integers and polynomials in a single variable over a box, and the connection among quantity idea and geometry*Many examples and countless numbers of difficulties are incorporated, in addition to tricks or whole suggestions for many of the problems*The exposition proceeds from the actual to the overall, usually delivering examples good earlier than a idea that comes with them; it contains blocks of difficulties that light up facets of the textual content and introduce extra topicsAdvanced Algebra offers its subject material in a forward-looking means that takes into consideration the ancient improvement of the topic.

Additional info for Torsion-free modules

Sample text

A, is a monomorphism. But then A' is a torsionless R-module, and hence by rank A' = rank A'". A' DEFINITIONS. can not be a is an isomorphism. An R-module will be called a universal injective R-module if it is injective and contains a copy of every simple R-module. J } the injective envelope of J. Then E() } is a universal injective R-module. An integral domain R is called a reflexive ring, if every torsionless R-module of finite rank is reflexive. By Theorem 27 this is equivalent to saying that every submodule of a finitely generated, torsion-free R-module is reflexive.

The following theorem presents one of the most useful aspects of h-local rings and we will make frequent use of it. THEOREM 24. R-module, then If R is an h-local domain, and C inj. dimR M over all maximal ideals of R. Thus CM, an where M ranges M gl. dim R = Proof. We will prove first that C s~r gl. dim RM • is an injective R-module if and only if CM is an injective RM-module for every maximal ideal M of R. Suppose that C is an injective R-module. If T is the torsion submodule of C, then C = T@ S, where S is torsion-free and divisible.

Therefore P is contained in only one maximal ideal of R. Let I be a nonzero ideal of R, and let T = R/I. a cotorsion R-module, we have, by assumption, that T"' Since T is TT TM. s Let M and N be distinct maximal ideals of R. N by Theorems 19 and 20 we have HomR(RM' T ) = O. SM= HomR(RM, ~~TN) ~ HomR(RM, TM)~ TM. N S - Let M T· since S is a cotorsion R-module, we have TT SM M 9! TT M TM 9! T, and hence S 9! R/I. Then Therefore, Thus 33 Now for N N (T )M ~ T N 0R Q "f M we have by Theorem 19 that = 0.

Download PDF sample

Rated 4.23 of 5 – based on 41 votes