subject of abstract algebra and no student should go through such a course without a good .. Theorem (De Morgan's Laws) Let A and B be sets. Then. 1. (A ∪ B)/ = A/ .. [7] Herstein, I. N. Abstract Algebra. 3rd ed. Wiley. Algebra. by Israel Nathan Herstein; Günter Drucks. Print book. German. by I N Herstein · Algebra moderna: Grupos, anillos, campos, teoría de Galois. present modern algebra as a lively branch of mathematics, having considerable . exercises be either stressed, de-emphasized, or omitted altogether.

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del Álgebra Abstracta desempeñan un papel cada vez más importante; por Dentro de la propia matemática el Álgebra Abstracta desempeña una dable. i. n. herstein. University of I approached rev1smg Topics in Algebra with a certain amount of two subsets A, B of S prove the De Morgan rules: (a) (A n B )'. Fourth Edition/Revision, (entitled Timeless Secrets of Health and Testing Your Mind/Body Response. 3 The Wonders Of O.

Includes index. ISBN alk. Algebra, Abstract. H47 '. No part of this book may bereproduced, in any form or by any means,without permission in writing from the publisher. Earlier editions and

Above I have described what I have added. What gave me greater difficulty about the revision was, perhaps, that which I have not added. I debated for a long time with myself whether or not to add a chapter on category theory and some elementary functors, whether or not to enlarge the material on modules substantially. After a great deal of thought and soulsearching, I decided not to do so. The book, as stands, has a certain concrete- ness about it with which this new material would not blend.

It could be made to blend, but this would require a complete reworking of the material Preface to the Second Edition v of the book and a complete change in its philosophy-something I did not want to do.

A mere addition of this new material, as an adjunct with no applications and no discernible goals, would have violated my guiding principle that all matters discussed should lead to some clearly defined objectives, to some highlight, to some exciting theorems.

Thus I decided to omit the additional topics. Many people wrote me about the first edition pointing out typographical mistakes or making suggestions on how to improve the book.

I should like to take this opportunity to thank them for their help and kindness. Preface to the First Edition The idea to write this book, and more important the desire to do so, is a direct outgrowth of a course I gave in the academic year at Cornell University. The class taking this course consisted, in large part, of the most gifted sophomores in mathematics at Cornell.

It was my desire to experiment by presenting to them material a little beyond that which is usually taught in algebra at the junior-senior level. I have aimed this book to be, both in content and degree of sophistication, about halfway between two great classics, A Survey of Algebra, by Birkhoff and MacLane, and Modern Algebra, by Van der Waerden. The last few years have seen marked changes in the instruction given in mathematics at the American universities. This change is most notable at the upper undergraduate and beginning graduate levels.

Topics that a few years ago were considered proper subject matter for semiadvanced graduate courses in algebra have filtered down to, and are being taught in, the very first course in abstract algebra.

Convinced that this filtration will continue and will become intensified in the next few years, I have put into this book, which is designed to be used as the student's first introduction to algebra, material which hitherto has been considered a little advanced for that stage of the game.

There is always a great danger when treating abstract ideas to introduce them too suddenly and without a sufficient base of examples to render them credible or natural. In order to try to mitigate this, I have tried to motivate the concepts beforehand and to illustrate them in concrete situations.

One of the most telling proofs of the worth of an abstract vii viii Preface to the First Edition concept is what it, and the results about it, tells us in familiar situations. In almost every chapter an attempt is made to bring out the significance of the general results by applying them to particular problems.

For instance, in the chapter on rings, the two-square theorem of Fermat is exhibited as a direct consequence of the theory developed for Euclidean rings. The subject matter chosen for discussion has been picked not only because it has become standard to present it at this level or because it is important in the whole general development but also with an eye to this "concreteness.

However, to appreciate this result for its own sake requires a great deal of hindsight and to see it used effectively would require too great a digression. True, one could develop the whole theory of dimension of a vector space as one of its corollaries, but, for the first time around, this seems like a much too fancy and unnatural approach to something so basic and down-to-earth.

Likewise, there is no mention of tensor products or related constructions.

There is so much time and opportunity to become abstract; why rush it at the beginning? A word about the problems. There are a great number of them. ISBN alk. Algebra, Abstract. H47 '.

No part of this book may bereproduced, in any form or by any means,without permission in writing from the publisher. Earlier editions and To Biska 5. We feel that one of the book's virtues isthe fact that it covers a big chunk of abstract algebra in a condensed and interestingway. At the same time, without trivializing the subject, it remains accessibleto most undergraduates.

We have, however, corrected minor errors, straightened out inconsistencies,clarified and expanded some proofs, and added a few examples.

To resolve the many typographical problems of the second edition,Prentice Hall has had the book completely retypeset-making it easier andmore pleasurable to read. It has been pointed out to us that some instructors would find it usefulto have the Symmetric Group Sn and the cycle notation available in Chapter2, in order to provide more examples of groups.

Rather than alter thearrangement of the contents, thereby disturbing the original balance, we suggestan alternate route through the material, which addresses this concern.

After Section 2. Thestudents might then go over several past examples of finite groups and explicitlyset up isomorphisms with subgroups of Sn- This exercise would be motivatedby Cayley's theorem, quoted in Section 2. At the same time, it wouldhave the beneficial result of making the students more comfortable with theconcept of an isomorphism.

The instructor could then weave in the varioussubgroups of the Symmetric Groups Sn as examples throughout the remain-ix 8.

If desired, one could even introduce Sections 3. Two changes in the format have been made since the first edition. First,a Symbol List has been included to facilitate keeping track of terminology. These serveas a vehicle to introduce concepts and simple arguments that relate in someimportant way to the discussion. As such, they should be read carefully. Finally, we take this opportunity to thank the many individuals whosecollective efforts have helped to improve this edition.

Barbara CortzenDavid J. Winter 9. For instance, the importance of the results and concepts of abstract algebraplay an ev;r more important role in physics, chemistry, and computer science,to cite a few such outside fields. In mathematics itself abstract algebra plays a dual role: that of a unifyinglink between disparate parts of mathematics and that of a research subjectwith a highly active life of its own.

It has been a fertile and rewarding researcharea both in the last years and at the present moment.

W e feel that one of the b o o k ' s virtues is the fact that it covers a big chunk of abstract algebra in a condensed and in- teresting way. A t the same time, without trivializing the subject, it remains ac- cessible to most undergraduates.

W e have, however, corrected m i n o r errors, straightened out inconsis- tencies, clarified and expanded some proofs, and added a few examples.

To resolve the many typographical problems of the second edition, Prentice Hall has had the b o o k completely retypeset—making it easier a n d m o r e pleasurable to read.

R a t h e r t h a n alter the arrangement of the contents, thereby disturbing the original balance, we sug- gest an alternate route through the material, which addresses this concern.

After Section 2. This exercise would be moti- vated by Cayley's theorem, q u o t e d in Section 2. At the same time, it would have the beneficial result of making the students more comfortable with the concept of an isomorphism. If desired, one could even introduce Sections 3. T w o changes in the format have been m a d e since the first edition.

First, a Symbol List has b e e n included to facilitate keeping track of terminology. T h e s e serve as a vehicle to introduce concepts and simple arguments that relate in some important way to the discussion. As such, they should be read carefully. Finally, we take this opportunity to thank the m a n y individuals whose collective efforts have helped to improve this edition.

And, of course, we thank George Lobell and Elaine W e t t e r a u , and others at Prentice Hall who have been most helpful.

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