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Serial Rings

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 520/5 ( 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 2012-12-06. 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.

Cyclic Modules and the Structure of Rings

2012-09-27 Mathematics
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Release : 2012-09-27
Genre : Mathematics
Kind : eBook
Book Rating : 545/5 ( reviews)

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Book Synopsis Cyclic Modules and the Structure of Rings by : S. K. Jain

Download or read book Cyclic Modules and the Structure of Rings written by S. K. Jain. This book was released on 2012-09-27. Available in PDF, EPUB and Kindle. Book excerpt: This unique and comprehensive volume provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. The finiteness conditions and homological properties are closely interrelated in the sense that either hypothesis induces the other in some form. This is the first book to bring all of this important material on the subject together. Over the last 25 years or more numerous mathematicians have investigated rings whose factor rings or factor modules have a finiteness condition or a homological property. They made important contributions leading to new directions and questions, which are listed at the end of each chapter for the benefit of future researchers. There is a wealth of material on the topic which is combined in this book, it contains more than 200 references and is not claimed to be exhaustive. This book will appeal to graduate students, researchers, and professionals in algebra with a knowledge of basic noncommutative ring theory, as well as module theory and homological algebra, equivalent to a one-year graduate course in the theory of rings and modules.

Algebras, Rings and Modules

2006-01-18 Mathematics
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Release : 2006-01-18
Genre : Mathematics
Kind : eBook
Book Rating : 919/5 ( reviews)

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Book Synopsis Algebras, Rings and Modules by : Michiel Hazewinkel

Download or read book Algebras, Rings and Modules written by Michiel Hazewinkel. This book was released on 2006-01-18. Available in PDF, EPUB and Kindle. Book excerpt: Accosiative rings and algebras are very interesting algebraic structures. In a strict sense, the theory of algebras (in particular, noncommutative algebras) originated fromasingleexample,namelythequaternions,createdbySirWilliamR.Hamilton in1843. Thiswasthe?rstexampleofanoncommutative”numbersystem”. During thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative algebras, began to bring some order into them and to single out certain types of algebras for special attention. Thus, low-dimensional algebras, division algebras, and commutative algebras, were classi?ed and characterized. The ?rst complete results in the structure theory of associative algebras over the real and complex ?elds were obtained by T.Molien, E.Cartan and G.Frobenius. Modern ring theory began when J.H.Wedderburn proved his celebrated cl- si?cation theorem for ?nite dimensional semisimple algebras over arbitrary ?elds. Twenty years later, E.Artin proved a structure theorem for rings satisfying both the ascending and descending chain condition which generalized Wedderburn structure theorem. The Wedderburn-Artin theorem has since become a corn- stone of noncommutative ring theory. The purpose of this book is to introduce the subject of the structure theory of associative rings. This book is addressed to a reader who wishes to learn this topic from the beginning to research level. We have tried to write a self-contained book which is intended to be a modern textbook on the structure theory of associative rings and related structures and will be accessible for independent study.

Index of patents

1927
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Release : 1927
Genre :
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Index of patents by :

Download or read book Index of patents written by . This book was released on 1927. Available in PDF, EPUB and Kindle. Book excerpt:

Algebras, Rings and Modules, Volume 2

2017-04-11 Mathematics
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Release : 2017-04-11
Genre : Mathematics
Kind : eBook
Book Rating : 876/5 ( reviews)

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Book Synopsis Algebras, Rings and Modules, Volume 2 by : Michiel Hazewinkel

Download or read book Algebras, Rings and Modules, Volume 2 written by Michiel Hazewinkel. This book was released on 2017-04-11. Available in PDF, EPUB and Kindle. Book excerpt: The theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This is the second volume of Algebras, Rings and Modules: Non-commutative Algebras and Rings by M. Hazewinkel and N. Gubarenis, a continuation stressing the more important recent results on advanced topics of the structural theory of associative algebras, rings and modules.

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