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13C05 Commutative algebra -- Theory of modules and ideals -- Structure, classification theorems

13-02 Commutative algebra -- Research exposition (monographs, survey articles) En-ligne: Springerlink | MathSciNet

Location | Call Number | Status | Date Due |
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Salle R | 06064-01 / 13 MAT (Browse Shelf) | Available |

Bibliogr. p. 153-154. Index

Let R be a commutative ring with identity and K its total ring of quotients. An R-module A is called Artinian if it satisfies the minimum condition for submodules. Note that an Artinian module may not satisfy the maximum condition; an easy example in case R is a discrete valuation ring is K/R. The author presents a structure theory for Artinian modules over a one-dimensional Noetherian Cohen-Macaulay ring. The local case is fundamental for the problem, and therefore we assume that R is a one-dimensional Cohen-Macaulay local ring.

One particular importance of the article lies in the development of a theory of divisible Artinian modules. For instance, the author defines two Artinian divisible modules to be equivalent to each other if each is a homomorphic image of the other, and then he generalizes the Jordan-Hölder theorem for Artinian modules. He also describes Artinian divisible modules in terms of a completion of R, and he gives a primary decomposition for Artinian divisible modules, and so forth.

The article begins with an introduction to basic notions on modules, completions, localizations and results developed by the author in some papers already published (Chapters I-IV). Then the author presents a new theory of Artinian divisible modules (Chapters V-XI). Then he observes the multiplicity theory and the theory of Gorenstein rings in the one-dimensional case (Chapters XII-XIII). In Chapter XIV he reconstructs the multiplicity theory of R and in the last chapter he shows that the existence of canonical ideals is decided by the divisible structure of K. (MathSciNet)

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