R-Modules & Homological Algebra

I studied completely the chapter 10 on Modules and the chapter 17 on an "Introduction to Homological Algebra" from the book of Dummit and Foote.

This book is not at all an easy reading for its giant, like the Bible of Abstract Algebra, but certainly it is the best I mean it brings everything: From Groups, to Rings, Modules, Fields, Field extensions, Galoi Theory, Commutative Algebra, Introduction To Algebraic Geometry and Homological Algebra.

I settled once and for all what a Module is. An R-Module M is a set M which is acted by left or right multiplication by an element of ring R, so that in the case you have M a abelian group and R a Field then you have the definition of a Vector Space. After going through the initial material of Modules, the chapter starts defining Chain Complexes and Cochain Complexes, these are row of mappings between the Modules such that the mappings are nilpotent. Then you define an exact sequence this is when the Image of the previous mapping is equal to the Kernel of the following mapping. Then it is shown that to each short exact sequence of modules there exists a long exact sequence of Cohomology groups. The Cohomology groups measure the obstruction of the initial sequence to be exact. They are defined as quotient spaces analogous and dual to Homology groups. Then The Functor HOM is defined. A functor just means a collection of functions as Categories are a collections of different types of sets. Turns out that there is a sequence of HOM functors which themselves are abelian groups or modules. This sequence can be extended by defining the Cohomology Groups EXT (from extensions they are defined as the quotient of the mapping of the image/kernel of the mappings in this sequence (The sequence of the Hom functors) Finally you can define a Homology Group which is a covariant functor called TOR which reverses the arrows in the cochain complexes turning it to a chain complex (TOR from Torsion) When the modules are projective or injective EXT=0. When the modules are Flat then TOR=0. Finally I learned what does it mean to say that an exact sequence splits, this means that the module in the middle of the short exact sequence can be written as a direct sum or direct (semi) product fo modules or groups respectively and of course there are conditions for that to happen.

I am very pleased cause I have settled once and for all Module Theory and I started seriously learning the first steps in Homological Algebra!

This is a superb textbook on algebra that is notable for its extremely clear and well-organized presentation. Development of different sections carefully builds on what went for, and running examples that gradually become more developed (for example, the quaternions as group, then as a ring, then various structural aspects) throughout. The terminology is completely standard, avoiding the temptation that some authors - or perhaps older texts - fall into of using bizarre terminology that is its author's favorite. The whole text has a very uniform, clear, well-architected feel to it: the sections stand on their own to the extent that they can, but also fit solidly into the rest of the presentation.

The presentation itself covers many topics which, taken together, make this an invaluable reference, for example group theory includes Burnside's theorem on solvability of certain finite groups (and at least mentions Feit-Thompson); ring theory includes a discussion of Gröbner bases; linear algebra includes symmetric and exterior algebras. A good introduction to algebraic geometry (the Nullstellensatz, localization, and some basic framework) is included. There is a solid introduction to representation theory via group rings and Wedderburn's theorem - an approach which is really more useful for applications than a pure group-theoretic introduction might have been.

Despite its broad coverage of topics, the book's development is extremely clear and easy-to-read. Because of the many examples and easy exercises, it is one of the most easy-to-understand texts I have seen. Every new idea is carefully defined and illustrated with multiple examples, proofs are very clear and painstaking. Indeed, given the methodical and well-motivated development, it's kind of a miracle that the text was able to include so much material; this is a testament to its excellent organization.

It's worth noting as well that the typographical layout is excellent for an advanced mathematics textbook. Rarely (if ever) have I read an advanced math textbook that was as well laid out typographically. There are, however, virtually no diagrams other than some subgroup lattices early on.

Caveats: there is little category theory - a choice I agree with at this level - and there is not much motivation of ideas outside math (or even outside algebra).

I would definitely recommend this book as an introduction, as a textbook, and as a reference. As a textbook, though, I might supplement it with some motivational notes on applications outside math (for example, coding theory, tiling, puzzles, and the like) and perhaps a few harder exercises, depending on the students.

Dummit and Foote contains just about everything an undergraduate ought to know about abstract algebra. In addition, it is written in a more user-friendly, down-to-earth fashion than, say, Lang's Algebra is.

The pro's have been discussed in other reviews and include: clear development of group, ring, and field theory; tons of exercises at the end of every chapter; numerous examples scattered around the text; sylow theorems (for group theory, imo, it's important, and not every algebra book does sylow stuff!); great introduction to exact sequences (useful if the reader is going into algebraic topology anytime soon. ugh!); galois theory is pretty clearly laid out; and, the third section of the book has some neat topics the reader can check out (which are, I think, commutative algebra, homological algebra, and representation theory introductions, as well as a small section on category theory at the very end).

The con's of D+F are the price (it's very expensive!), the binding (it's horrible!), and some of the sections are much harder than others and D+F doesn't do as well a job at explaining them as in many of the other sections (the tensors section sticks out in my head, and they wait something like 100 pages to explain "tricks" for figuring out the structure of finite groups after explaining some of the sylow stuff (eg., they wait to tell the reader about how to "pin small groups against one-another" and to make use of the sylow n! trick). Also, D+F introduce modules before vector spaces which I have mixed feelings about --- as a student who's already taken an algebra class, I love the "flow" of the lessons; as a student who remembers what it was like to try to imagine what modules "looked like", it makes me cringe to think that they didn't introduce vector spaces first.

Overall, wonderful book. One of my favorites of all time. DEFINITELY have it, and if you study from it, you may feel more comfortable supplimenting it with Herstein's Algebra, Artin's Algebra (which are just as hard) or Fraleigh's Abstract Algebra, Gallian's Abstract Algebra, or Rotman's Abstract Algebra (which are much, much easier).

I used this book in an advanced undergraduate/master's level algebra course. I mostly used this book for exercises but on the occasion that I read the chapters they were friendly enough and readily digestible. The real pro about this book is the high number of non-trivial exercises. There are more than enough to get a good feeling out of any topic in the book and problems range from relatively straightforward to moderately sophisticated. There are also a good deal of computational problems of varying difficulty and many worked out examples in the book. As for the book's organization: the sections on group theory don't follow the best pattern, to me, but there is nothing seriously wrong with it. Sometimes D+F's proofs can be excessively wordy or miss proper quantifiers, but for the most part everything is easy enough to follow. D+F's exposition is decent, but I've read better. I seldom used this book to teach myself a subject without first learning about it in lecture, so I can't comment too much here.

The price is fairly high, but the book is huge. There is plenty of material here and it's arguably one of the few math books worth the list price. Nevertheless, you'd be better off finding a used copy as there are most likely many people who could not handle the subject and returned the book. If you're serious about learning algebra this is a must-have book. It will prepare you well for more advanced studies. There also seems to be a problem with the binding as my book and many of my classmates had problems with the cover ripping or spine splitting.

Has nearly everything you need or want to know (and even stuff you didn't want to know) about undergraduate abstract algebra --- without being too abstract.

The book is comparatively wordy compared to other algebra texts (this book is twice the length of, say, hungerford's algebra while covering the same content); this is good for learning purposes (as things are explained in detail), but the book is kinda too big to be a handy reference (compared to lang's algebra for instance).

The only real problems one might have with this book may be as follows: (1) it's not categorical enough --- universal properties are stated and proved in the text (e.g., with free modules and tensor products), yet the book only devotes 7 pages to category theory in the appendix, and (2) chapters 15-16 (commutative algebra/introductory algebraic geometery) are (comparatively) a trainwreck and very difficult to learn from. But the theory of groups/rings/modules/fields (the basic algebra course) is excellent so that could be generally overlooked.

the c hard cover looks sturdy; 2 or 3 soft covers fell apart; time will tell(given that i live long enough)

I am taking a graduate level algebra course, and I've found many of my previous texts to be insufficient as a reference when I need to bail myself out of a jam. Abstract Algebra is a wonderful (and it is still one of my favorites) text to introduce yourself to the subject, but it lacks as a reference. My friend suggested that I pick this book up because it covers pretty much anything you'd need to know at an undergraduate level, and much more. Well, he was right!

Dummit and Foote's exposition is very clear. The book is loaded with examples and the authors don't shy away from details, so you're not left scratching your head when you need information quickly. Most of their notation is fairly standard, and when it's not, the notation they do use is easily understood (and they have a reference table in the front!)

Ultimately, if you plan on studying mathematics at the graduate level, this book could be the only undergraduate algebra reference text you'll ever need since it's so comprehensive. This would, in my opinion, also serve as a very good 1 year course on Algebra for undergrads, as there are many different routes one can take through it, and it is very much self-contained.

I never took abstract algebra in college but the writing style, presentation, and wealth of informative examples made this text perfect for self study. It really is a classic for me now in my library and I refer to it almost daily now. The hardest part is sticking with the initial group theory which seemed odd, but by the time you get to Gallois theory it’s just remarkable how far the same few “techniques” can get you across all these various objects. It’s just beautiful. I cannot recommend this book enough and honestly, you don’t need any calculus at all or even analysis, just logic and set theory could get you started for the most part. I’m confused why abstract algebra is taught so late the more I’ve pondered over things in this book and realized how much more sense everything makes thinking about coordinate systems and mappings which before seemed contrived or very hazy.

I did supplement the book with Benedict Gross’ Harvard lectures which really helped in some areas but overall I could have got by without.

There are not many great books out there, but this one is a real gem. I can whole heartedly recommend this book.

If you have a background in physics, like I do, it will help you with the more mathematical foundations of quantum mechanics.

This book combined with J.J. Sakurai's Advanced Quantum Mechanics is a great for studying.

Dummit and Foote is a classic, and probably for a good reason.

Recebi hoje 31/03/2024. Excelente impressão e paginação de primeira qualidade.

Heel duidelijk uitgelegd van het begin tot ver gevorderde algebra, gebruikt voor de universiteit richting wiskunde en duidelijker dan de uitleg van sommige profesoren in de les.

Jammer dat het niet in plastic vast zat want nu waren de eerste 30 bladeren met een vouw erin door te kunnen bewegen bij het transport.

The purchase actually arrived ahead of time and the book was exactly as advertised.