From this humble origin, it flourished into a field of study in its own right of an astonishing richness and interest. Nowadays, one has to specialize in an area of this vast field in order to be able to master its wealth of results and come up with worthwhile contributions. One of the major areas of the field of Commutative Ring Theory is the study of non-Noetherian rings.

The last ten years have seen a lively flurry of activity in this area, including: a large number of conferences and special sections at national and international meetings dedicated to presenting its results, an abundance of articles in scientific journals, and a substantial number of books capturing some of its topics.

This rapid growth, and the occasion of the new Millennium, prompted us to embark on a project aimed at presenting an overview of the recent research in the area.

With this in mind, we invited many of the most prominent researchers in Non-Noetherian Commutative Ring Theory to write expository articles representing the most recent topics of research in this area. Product Details Table of Contents. Table of Contents Preface.

Mori Domains; V. Cahen, J. Half-Factorial Domains, a Survey; S. Chapman, J. Chapman, et al.

## Noetherian ring

Localizing Systems and Semistar Operations; M. Fontana, J. Ideal Theory in Pullbacks; S. Gabelli, E. Commutative Rings of Dimension 0; R. Heinzer, M. Connecting Trace Properties; J. Huckaba, I. T-Closedness; G. Picavet, M.

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E-rings and Related Structures; C. Prime Ideals and Decompositions of Modules; R. Wiegand, S. Putting t-Invertibility to Use; M. Chapman, S. Show More.

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Choosing a basis, we can describe the same ring R as. A unique factorization domain is not necessarily a Noetherian ring.

It does satisfy a weaker condition: the ascending chain condition on principal ideals. A valuation ring is not Noetherian unless it is a principal ideal domain.

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It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian. In the ring Z of integers, an arbitrary ideal is of the form n for some integer n where n denotes the set of all integer multiples of n. This is referred to as a primary decomposition of the ideal n. In Z , the primary ideals are precisely the ideals of the form p e where p is prime and e is a positive integer.

Thus, a primary decomposition of n corresponds to representing n as the intersection of finitely many primary ideals.

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## Noetherian ring

For all of the above reasons, the following theorem, referred to as the Lasker—Noether theorem , may be seen as a certain generalization of the fundamental theorem of arithmetic:. Lasker-Noether Theorem. Let R be a commutative Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:.

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Furthermore, if:. From Wikipedia, the free encyclopedia. Algebraic structures Group -like. Ring -like. Lattice -like. Module -like. Module Group with operators Vector space. Algebra -like. Main article: Lasker—Noether theorem.