By Eben Matlis
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.
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Additional info for Torsion-free modules
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.