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Is it possible to learn ring theory if one is familiar with but not good at group theory?

Background: I’m using Dummit and Foote's Abstract Algebra, and I am an undergrad.

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Rings have more structure than groups due to the presence of a multiplication operation, so there are more structural results coming from the interplay between addition and multiplication. Just pick up any book on commutative algebra, and you will be amazed by the wealth of structural results for rings. You get even more of such results in field theory, such as those encapsulated by Galois theory. However, once you go down to the level of groups, there is less structure to play with, so results are harder to come by. We still have powerful theorems like Sylow’s Theorems, though. –  Haskell Curry Jan 7 '13 at 21:57
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Hence, it is possible to learn ring theory before being proficient in group theory, especially if you like a friendlier environment that contains more theorems that you can add to your toolbox. –  Haskell Curry Jan 7 '13 at 21:59
    
At Warsaw University it was very common to start the "Algebra 101" class with rings instead of groups. Indeed, the rings one is interested at the beginning are mostly the polynomial and number rings which most students felt pretty comfortable and the theorems about them didn't go very deep. The "group part" of the course ended with the Sylow theorems, which were much more difficult. –  Piotr Pstrągowski Jan 8 '13 at 18:42

6 Answers 6

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Is it possible to learn ring theory if one's familiar, but not good at group theory?

Absolutely yes; your study of Dummit and Foote should give you enough of a foundation in "group theory" to successfully study "ring theory" in greater depth. So it is indeed possible to learn ring theory before being fully proficient in group theory.

But I also want to add:

It is far too early for you to conclude you are not good at group theory!
You're an undergrad, as you point out, and your experience when first encountering any new theory or topic is not really sufficient evidence to determine how "good" you are at it, especially if you're making that judgment based on your progress through one book's coverage of groups.

Every serious student of math encounters a wall at one point or another (and for most of us, many many times). Sometimes the things that first "trip us up" are precisely the things that end up fascinating us.

Furthermore, try not to judge your mastery of any topic (e.g. group theory) based on its presentation in only one text. (In my humble opinion) Dummit and Foote's text is not all that great in covering groups.

So explore a bit by supplementing Dummit and Foote's coverage of groups with other resources: e.g., find a text or two that approach groups differently:

  • Artin's Algebra covers groups well; see especially chapter 2.

  • Or - if you're really struggling with groups - try looking at Fraleigh's A First Course in Algebra, which does great with introducing groups and motivating the material. The book is written in a way that is very readable, intuitive, and includes a lot of examples.

  • If you can't access one of the above texts through a library, and are looking to limit expense, J.S. Milne has a nice site for course-notes, including a ~$140$-page pdf on Group Theory.

So try not to "write off" group theory quite yet; you may find you like group theory, after all!

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@Asaf - hehehehe! Almost...! –  amWhy Jan 13 '13 at 0:21
    
@Thanks for noticing, Asaf! –  amWhy Jan 13 '13 at 0:22

It is certainly possible to study rings before groups. In fact, this is the approach taken in say Shifrin's Abstract Algebra as rings are perhaps more natural than groups to some.

On the other hand, the study of rings will involve some group-theoretic results, but these can always be picked up on the go.

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Rings also appear before Groups in P.J. Cameron's Introduction to Algebra. –  Clive Newstead Jan 7 '13 at 18:12
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And in Tom Hungerford’s undergraduate text. –  Brian M. Scott Jan 7 '13 at 18:27

Personally, I think "Dummit & Foote" is excellent for ring theory. And much more accessible for a beginner than groups. (Dummit wrote rings, Foote groups.)

To build more confidence in groups, the chapter in Artin's "Algebra" (go for the 2nd edition) on groups is very well written and can give a very intuitive understanding of what are key points.

You can also watch the specific videos on groups in this excellent lecture series by Benedict Gross at Harvard, that follow Artin. You will most likely want to watch the ones on rings as you study that topic.

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

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Ring theory works with commutative groups, certainly in the early stages - fields are commutative, polynomial rings are commutative - you can do a fair amount without having to deal with the non-commutative case (think of the large text books on commutative algebra). This is a major simplification.

Also, as others have mentioned, there are some very familiar canonical examples of rings.

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I think this is the topic in which courses differ the most... Some courses start with rings because it's the historical way, starting with Galois works on rings based on his tudies about polinomials. Groups appeared after that. Some other courses start with groups going not the historical way, but the formal way, so when you get to rings you can use the knowledge you have about groups. So, as they have said, absolutely you can study ring theory with little knowledge about groups, as many textbooks dont presume any knowledge.

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I must say I don't see why you are interested in ring theory if you are not good at group theory ; you don't give interest in a subject just because it looks cool, but because you want to be able to use it in some way. If you just want to read about it, you can, whether you are good or not, it's not a problem. Your eyes will still open and you will still be able to read the book! The issue is when you will want to do exercises ; group theory and ring theory are theories among the algebraic theories, so their nature is very similar ; it is very likely that being good/bad at understanding one will imply being good/bad at understanding the other, although it is possible that you feel a little bit more comfortable working with objects with a little bit more structure than groups.

As amWhy points out, we interpret your "not being good at group theory" as "you're not comfortable with it". You might turn out to feel comfortable with it later and be very good at it.

If this is because you want to read about a ring theory course you must take before you finished your group theory class or something like that, I think it is a bad idea ; the Dummit & Foote book is a good book to use as a reference and is full of relevant exercises, I am a fan of it myself, but I must say I did not begin learning group theory using this one, it lacks a little bit intuition. For books that build intuition on groups in a more relevant way, I could provide you a reference.

I'll add that when one begins ring theory, the first examples of rings differ a lot from the group-theoretic ones, mostly by the fact that they are infinite. A lot of the elementary group theory focuses on finite groups because they are easier to study ; in ring theory, the basic examples would come from subrings of fields ($\Bbb Z$ or $n\Bbb Z$ would be an example), number rings or function rings, and their quotients. This gives them a quite different feeling from the elementary examples in group theory such as $\Bbb Z/n \Bbb Z$, $S_n$, $A_n$, $D_n$, etc.

To sum things up : give it a try and see what happens, if you have the time to do so.

Hope that helps,

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Actually, I do take an interest in a subject because it looks interesting; I don’t care whether I can use it in some way (other than for my own enjoyment). –  Brian M. Scott Jan 7 '13 at 18:26
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And actually, one can be interested in things that one doesn't feel good at. "Not being good at" something is subjective, and subject to change (as one learns). At any rate, "interest in" isn't necessarily positively correlate with "good at" and "not good at" doesn't necessarily correlated with "lack of interest in." Some of us are prone develop interest in tackling that which we feel weak in. –  amWhy Jan 8 '13 at 0:59
    
@Brian : If you would take interest in every subject that looks interesting, you would have time for nothing ; there are LOTS of things of interest, even to you, in this world. You mostly give interest in things that relate to you in some way because you can use them or see what happens when... "etc.". You don't just start doing biology even though you're not good in biology and will never use biology in your life. It's one thing to read about it on a Facebook post and it's another one to take a course. –  Patrick Da Silva Jan 10 '13 at 2:52
    
And it's not because Brian posted a very relevant comment that I deserve a downvote ; it's a soft question, discussion is normal. I believe I did answer the question. –  Patrick Da Silva Jan 10 '13 at 2:52
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Yes, I’m interested in a great many things, more than I really have time for. That’s why I have to set some of my interests aside for a while from time to time. But the fact remains that I do get interested in things for no obvious reason beyond the fact that I happen to encounter them. Learning Old Norse and reading sagas in the original language serve no other purpose. Studying medieval onomastics serves no other purpose. I do these things because I find them interesting and enjoyable, no more, no less. I occasionally play the recorder for the same reason, and I’n not good at that. –  Brian M. Scott Jan 10 '13 at 16:21

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