Topics in Commutative Ring Theory - Hardcover

Watkins, John J.

Hardcover

ISBN 10: 0691127484 ISBN 13: 9780691127484

Verlag: Princeton University Press, 2007

Inhaltsangabe

Topics in Commutative Ring Theory is a textbook for advanced undergraduate students as well as graduate students and mathematicians seeking an accessible introduction to this fascinating area of abstract algebra.


Commutative ring theory arose more than a century ago to address questions in geometry and number theory. A commutative ring is a set-such as the integers, complex numbers, or polynomials with real coefficients--with two operations, addition and multiplication. Starting from this simple definition, John Watkins guides readers from basic concepts to Noetherian rings-one of the most important classes of commutative rings--and beyond to the frontiers of current research in the field. Each chapter includes problems that encourage active reading--routine exercises as well as problems that build technical skills and reinforce new concepts. The final chapter is devoted to new computational techniques now available through computers. Careful to avoid intimidating theorems and proofs whenever possible, Watkins emphasizes the historical roots of the subject, like the role of commutative rings in Fermat's last theorem. He leads readers into unexpected territory with discussions on rings of continuous functions and the set-theoretic foundations of mathematics.


Written by an award-winning teacher, this is the first introductory textbook to require no prior knowledge of ring theory to get started. Refreshingly informal without ever sacrificing mathematical rigor, Topics in Commutative Ring Theory is an ideal resource for anyone seeking entry into this stimulating field of study.

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John J. Watkins

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"A very elementary introduction to commutative ring theory, suitable for undergraduates with little background. It is written with great care, in a conversational and engaging style that I think will appeal to students. Essentially every detail is made explicit, and readers are admonished to beware typical pitfalls. The book is also peppered with very nice detours into the history of mathematics."--Karen Smith, University of Michigan

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"A very elementary introduction to commutative ring theory, suitable for undergraduates with little background. It is written with great care, in a conversational and engaging style that I think will appeal to students. Essentially every detail is made explicit, and readers are admonished to beware typical pitfalls. The book is also peppered with very nice detours into the history of mathematics."--Karen Smith, University of Michigan

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Topics in Commutative Ring Theory

By John J. Watkins

Princeton University Press

Copyright © 2007 Princeton University
All right reserved.
ISBN: 978-0-691-12748-4

Chapter One

Rings and Subrings

The Notion of a Ring

In 1888 - when he was only 26 years old - David Hilbert stunned the mathematical world by solving the main outstanding problem in what was then called invariant theory. The question that Hilbert settled had become known as Gordan's Problem, for it was Paul Gordan who, 20 years earlier, had shown that binary forms have a finite basis. Gordan's proof was long and laboriously computational; there seemed little hope of extending it to ternary forms, and even less of going beyond. We will not take the time here to explore any of the details of Gordan's problem or even the nature of invariant theory (and you shouldn't be at all concerned if you don't have the foggiest idea what binary or ternary forms are or what a basis is), but Hilbert - in a single brilliant stroke - proved that there is in fact a finite basis for all invariants, no matter how high the degree.

The structure of Hilbert's proof is really quite simple and is worth looking at (again we will not worry at all about most of the details). First, Hilbert showed that if a ring R has a certain property P, then the ring of polynomials in a single variable x with coefficients from the ring R also has that same property P. (Today we would say that P is the property that any ideal is finitely generated, but that is getting well ahead of our story.) We will use the convenient notation R[x] to represent this ring of polynomials in x with coefficients from R, and so we can summarize Hilbert's first step as

if a ring R has property P, then so does the ring R[x].

Next, Hilbert wanted to show that the ring of polynomials in two variables x and y with coefficients from the ring R also has property P. We represent this ring of polynomials in two variables by R[x, y]. These polynomials are just like ordinary polynomials we are used to such as [x.sup.2] + 4 and 2[y.sup.2] - y + 3, except that now we can have the two variables x and y mixed together in a single polynomial. An example of such a polynomial is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where in this case the coefficients are all integers. Hilbert saw that this ring of polynomials R[x, y] in two variables x and y with coefficients from the ring R can be thought of as the ring of polynomials in a single variable y with coefficients from the ring of polynomials in x. For example, we can write the above polynomial in two ways, depending on how we choose to group the terms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the left the polynomial is written as a polynomial in the ring R[x,y], whereas on the right it is written as a polynomial in one variable y where the coefficients are themselves polynomials in x. Our notation for this latter ring is (R[x] [y] - or more simply R[x][y] - emphasizing the fact that the coefficients are now polynomials in the ring R[x].

Using this simple idea, Hilbert concluded that the ring of polynomials R[x, y] also has property P. His argument went like this: since the ring R has property P, so does the ring R[x]; but then since the ring R[x] has property P, so does the ring R[x][y]; and, as we have just seen, this latter ring is really the same as the ring R[x, y].

In this way, by adding one variable at a time, Hilbert showed that the polynomial ring in any finite number of variables has property P. For example, we could now conclude that the ring R[x, y, z] has property P since this ring is the same as the ring R[x, y][z] and we have just argued that R[x, y] has property P. The key to Hilbert's argument, then, is to verify his very first step - namely, that if a ring R has property i, then so does the polynomial ring with coefficients from R.

Now Hilbert did this not by explicitly constructing a basis (as Gordan had done for the binary case), but rather - and this is the brilliant part of his proof - by showing that if there were no finite basis, then a contradiction arises. Therefore, there must be a finite basis after all! Nowadays, we are very comfortable with such a proof by contradiction, but Hilbert had used this technique in a new way: he had proved the existence of something without actually constructing it. This existence proof did not meet with universal favor in the mathematical climate of his day. In fact, Gordan - hardly an impartial observer - chose this time to issue one of the most memorable lines in all of mathematics: "Das is nicht Mathematik. Das ist Theologie." It was not until four years later, when Hilbert was able to use the existence of a finite basis to show how such a basis could actually be constructed, that Gordan conceded: "I have convinced myself that theology also has its advantages."

At the heart of Hilbert's proof - and the attendant controversy - lies the abstract notion of a ring, though it would be several years until Hilbert would actually provide us with the term ring (or Zahlring - literally, number ring) which we now use today. The idea is that, for instance, although polynomials certainly differ in many obvious ways from integers, there are ways in which polynomials and integers are similar: for example, you can add or multiply integers and you can also add or multiply polynomials. It is the differences between integers and polynomials that most of us notice first, but Hilbert focused instead on their similarities. So, the idea behind the notion of a ring is that integers, rationals, reals, complex numbers, polynomials with complex coefficients, and continuous functions, as different as all of these systems may appear to us, all share certain characteristics. It is these shared underlying characteristics which provide the basis for the following unifying axioms and our definition of a ring, for it is the abstract notion of a ring that so elegantly captures the essence of what these familiar mathematical systems share in their behavior.

The Definition of a Ring

Before we actually define a ring, let us talk a bit about what a ring is. Quite simply it is a set of elements (typically a set of numbers of some kind, or perhaps a set consisting of a particular type of function) together with two operations on those elements called addition and multiplication. It is very important to think of a ring as a single object consisting of both the underlying set and the two operations, and not just as a set by itself. Furthermore, these operations will need to behave the way we expect them to behave. For example, if a and b are two elements in a ring, we expect a + b and b + a to be equal, or we expect a + 0 to equal a, or we expect a(a + b) to equal [a.sup.2] + ab. We have these expectations no matter whether a and b are numbers, or polynomials, or matrices.

Let us look at some specific examples of rings. In each case, note that we present both a set and two operations on that set in order to describe the ring.

Example 1

Certainly the...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2007
• Sprache: Englisch
• ISBN 10: 0691127484
• ISBN 13: 9780691127484
• Einband: Gebundene Ausgabe
• Auflage: 1
• Anzahl der Seiten: 228
• Kontakt zum Hersteller: preigu GmbH & Co. KG; mail@preigu.de; ; Lengericher Landstr. 19; Osnabrück; Niedersachsen; 49078; Deutschland