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Krull Dimension

1973 Associative rings
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Release : 1973
Genre : Associative rings
Kind : eBook
Book Rating : 333/5 ( reviews)

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Book Synopsis Krull Dimension by : Robert Gordon

Download or read book Krull Dimension written by Robert Gordon. This book was released on 1973. Available in PDF, EPUB and Kindle. Book excerpt: This memoir is devoted to the study of Krull dimensions of modules over (not necessarily commutative) rings with identity. Although not all modules have a Krull dimension, Noetherian modules always do. There has been much interest in the Krull dimension of noncommutative Noetherian rings and their modules. In this memoir, as well as providing an exposition of much of what is known about Krull dimension, we study those rings and modules, not necessarily Noetherian, which have a Krull dimension. The results we obtain show that merely having a Krull dimension is a stringent condition with many Noetherian-like consequences. We expect these results, and the methods involved, to have applications to Noetherian and related rings.

Ideals and Reality

2004-11-24 Mathematics
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Release : 2004-11-24
Genre : Mathematics
Kind : eBook
Book Rating : 328/5 ( reviews)

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Book Synopsis Ideals and Reality by : Friedrich Ischebeck

Download or read book Ideals and Reality written by Friedrich Ischebeck. This book was released on 2004-11-24. Available in PDF, EPUB and Kindle. Book excerpt: Besides giving an introduction to Commutative Algebra - the theory of c- mutative rings - this book is devoted to the study of projective modules and the minimal number of generators of modules and ideals. The notion of a module over a ring R is a generalization of that of a vector space over a field k. The axioms are identical. But whereas every vector space possesses a basis, a module need not always have one. Modules possessing a basis are called free. So a finitely generated free R-module is of the form Rn for some n E IN, equipped with the usual operations. A module is called p- jective, iff it is a direct summand of a free one. Especially a finitely generated R-module P is projective iff there is an R-module Q with P @ Q S Rn for some n. Remarkably enough there do exist nonfree projective modules. Even there are nonfree P such that P @ Rm S Rn for some m and n. Modules P having the latter property are called stably free. On the other hand there are many rings, all of whose projective modules are free, e. g. local rings and principal ideal domains. (A commutative ring is called local iff it has exactly one maximal ideal. ) For two decades it was a challenging problem whether every projective module over the polynomial ring k[X1,. . .

Dimensions of Ring Theory

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 357/5 ( reviews)

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Book Synopsis Dimensions of Ring Theory by : C. Nastasescu

Download or read book Dimensions of Ring Theory written by C. Nastasescu. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of s9phistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Commutative Ring Theory and Applications

2017-07-27 Mathematics
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Release : 2017-07-27
Genre : Mathematics
Kind : eBook
Book Rating : 627/5 ( reviews)

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Book Synopsis Commutative Ring Theory and Applications by : Marco Fontana

Download or read book Commutative Ring Theory and Applications written by Marco Fontana. This book was released on 2017-07-27. Available in PDF, EPUB and Kindle. Book excerpt: Featuring presentations from the Fourth International Conference on Commutative Algebra held in Fez, Morocco, this reference presents trends in the growing area of commutative algebra. With contributions from nearly 50 internationally renowned researchers, the book emphasizes innovative applications and connections to algebraic number theory, geome

Serial Rings

2001-08-31 Mathematics
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Release : 2001-08-31
Genre : Mathematics
Kind : eBook
Book Rating : 878/5 ( reviews)

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Book Synopsis Serial Rings by : G. Puninski

Download or read book Serial Rings written by G. Puninski. This book was released on 2001-08-31. Available in PDF, EPUB and Kindle. Book excerpt: The main theme in classical ring theory is the structure theory of rings of a particular kind. For example, no one text book in ring theory could miss the Wedderburn-Artin theorem, which says that a ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of full matrix rings over skew fields. This is an example of a finiteness condition which, at least historically, has dominated in ring theory. Ifwe would like to consider a requirement of a lattice-theoretical type, other than being Artinian or Noetherian, the most natural is uni-seriality. Here a module M is called uni-serial if its lattice of submodules is a chain, and a ring R is uni-serial if both RR and RR are uni-serial modules. The class of uni-serial rings includes commutative valuation rings and closed under homomorphic images. But it is not closed under direct sums nor with respect to Morita equivalence: a matrix ring over a uni-serial ring is not uni-serial. There is a class of rings which is very close to uni-serial but closed under the constructions just mentioned: serial rings. A ring R is called serial if RR and RR is a direct sum (necessarily finite) of uni-serial modules. Amongst others this class includes triangular matrix rings over a skew field. Also if F is a finite field of characteristic p and G is a finite group with a cyclic normal p-Sylow subgroup, then the group ring FG is serial.

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