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I am of, and I would like to retain, a mindset that mathematics does not have to have numbers as the central object of interest. With that in mind, I have done a fair amount of self-study on topics in undergrad modern algebra, looking for examples to show this is the case.

For objects like groups, it is often stated explicitly, at some point, that group elements can be though of as automorphisms on a particular set. I find this pretty interesting, that numbers do not come into play at all here. This makes groups pretty useful for problems that have symmetries.

However, for objects like fields and rings, it seems that their application often ends up dealing with numbers somehow. I'm aware there are probably a handful of cases where rings and fields do not show up as numbers, but those tend to be side-cases that are not of main study. I'm only vaguely aware of Galois theory and it's uses for fields as extensions of groups, so perhaps that is the direction I should follow there.

But as for rings, aside from the example I just gave, I still come up a bit short on any powerful or broad theorems that make use of those two operations but do not involve numbers in some fundamental way. Perhaps math.stackexchange can prove me wrong?

P.S. Answers with really good links to books or other resources will at least get an upvote.

An update for those who pointed out my lack of precision.

So admittedly, the choice of words was bad. I did use the phrase "involve numbers in some fundamental way", which, in my mind, means "does not use facts about numbers". Perhaps I can rephrase it better.

Suppose I have a group (G, *). There are many examples of groups if I want to let G be the integers, rationals, reals, complex, etc, where the group operation (*) can be addition, multiplication, or some other operation.

However, each of these are "numerical" examples. There are groups which do not behave like numbers, if we let G be the set of all bijections on a set S, and let (*) be the composition, we may not be able to find a group where the set S is a collection of numbers and (*) is a "numerical" operation on them. Our group operation (*) may end just being some random rearrangement of numbers that does not take advantage of the properties of numbers.

In this way, and as others often point out, all groups are just specific cases of bijections & composition.

Now suppose I have a ring (R, +, *). The ring of functions on integers, rationals, reals, etc. takes advantage of the properties of numbers when defining (+) and (*). As would the ring of polynomials, or the ring of matrices. Each of these examples of rings, while the set R is not a set of numbers, ends up being an abstraction of numbers, and the operations (+) and (*) make use of this fact.

An example of what I mean

To give a example (counterexample?) of what I mean, try Boolean rings. The operations (+) and (*) are isomorphic to the logical operations "xor" and "and". There is also examples of lattices with join and meet. One could try to squeeze in natural numbers into this with lcd and gcd, but these properties are not unique to numbers so they are not exactly "numerical".

If you want to add more structure, say an additive and multiplicative identity, you can use Stone's Representation Theorem to show it is isomorphic to a collection of sets with symmetric difference and intersection.

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One of the standard examples of a ring is the polynomial ring (or more generally, rings of functions on some space; for polynomial rings, this is affine space). For instance, one of the first results you prove in commutative algebra is that if a ring $R$ is noetherian (an extremely useful finiteness condition that crops up all the time), then so is the polynomial ring $R[X]$. –  Akhil Mathew Oct 11 '10 at 13:01
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I disagree with the premise of the question. Many of the most important examples of rings have nothing to do with numbers and are certainly "of main study." –  Qiaochu Yuan Oct 11 '10 at 14:55
    
Galois theory is a very beautiful field, but it's not the way to go if you want to avoid numbers. It was invented to study roots of polynomials in $\mathbb{Z}[X]$, and though its Fundamental Theorem is nice and abstract, almost anything you'd use it for is going to involve polynomials over a finite field or an extension of $\mathbb{Z}$ or $\mathbb{Q}$. –  Paul VanKoughnett Oct 11 '10 at 16:32
    
Wow, big edit. I really don't think the question is well-defined, though. For instance, how "numerical" is $\mathbb{Z}_n$? What about $\mathbb{F}_p$? As you yourself noted, any finite group embeds in a symmetric group -- if we choose to view it as this embedding, does that make it less "numerical"? Two interesting examples of rings are the ring of integrable functions with convolution as the multiplication, and the group ring $\mathbb{Z}[G]$. Though they both depend on the additive properties of $\mathbb{Z}$, I wouldn't consider them numerical. Would you? –  Paul VanKoughnett Oct 12 '10 at 21:01
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Dear robinhoode, As an easier question, you might try to ask yourself whether you can construct many natural examples of abelian groups that do not involve numbers in some way. (The point is that groups like symmetric groups, or other automorphism groups, tend to be non-commutative; abelian groups come up much more naturally in contexts where the group operation is obtained in some way from the addition of numbers.) –  Matt E Oct 13 '10 at 3:09

6 Answers 6

up vote 9 down vote accepted

I think this is a legitimate observation -- it is possible to do lots of group theory without knowing much about integers or real numbers, but these number systems seem to come up immediately whenever one is dealing with rings.

It seems to me that the main reason is that every ring has a characteristic, and therefore a ring with unity necessarily includes a copy of either the integers or some $\mathbb{Z}_n$. Furthermore, any element of a finitely-generated ring can be written as a (possibly noncommutative) polynomial in the generators, where the coefficients of the polynomial are integers (or perhaps some elements of $\mathbb{Z}_n$). Adding and multiplying these polynomials inherently requires addition and multiplication of integers.

Once you get past this basic "number-ness", there are many rings that are otherwise unrelated to either integers or real numbers. For example, it's possible to develop the theory of polynomial rings quite a bit without using numbers for anything other than coefficients. The same goes for group rings, whose elements are hardly more number-like than typical group elements. For a more sophisticated example, the elements of the (integer) cohomology ring of a topological space are essentially geometric objects, not numbers.

Fields are the same way. Every field inherently contains either the rational numbers $\mathbb{Q}$ or a prime field $\mathbb{Z}_p$, but aside from that one could argue that many fields have very little to do with numbers.

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You can think of a group (e.g., via Cayley's Theorem) as a collection of bijections that contains the identity and is closed under composition and inverses. So groups are objects that "act on" arbitrary sets (as bijections). That seems to be what you are talking about when you talk about group elements being "automorphisms of a particular set".

As Jack Schmidt pointed out in MathOverflow, every associative unital ring is a subring of the endomorphism ring of its underlying additive group. That is, you can think of any ring as being a collection of group endomorphisms of some abelian group, with the ring addition corresponding to the usual addition of endomorphisms (inherited from the group), and ring multiplication corresponding to composition of functions. This closely parallels the view of "groups as collections of bijections on a set": rings are "collections of endomorphisms of a group". If the abelian group has nothing to do with numbers, then the corresponding ring will also have "nothing to do with numbers".

If you think of groups as objects that act on sets as bijections, then you can think of (associative unital) rings as objects that act on abelian groups as endomorphisms.

I don't know if this answer is satisfying, but it seems to me to be along the lines you are inquiring.

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I was kinda hoping for a bit more, but this is probably the best answer so far. Thanks. –  Robin Hoode Oct 15 '10 at 0:15
    
I've once heard it claimed that rings are not important because they're rings, but because they act on modules. –  Hurkyl Sep 12 '13 at 9:14

If $F$ is a field, then $F[X]$ is a PID. If $D$ is a UFD, then so is $D[X]$. These are broad and powerful theorems.

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Rings of "functions" (vs. numbers) are heavily studied - whether formally or truly functionally. Even in a first algebra course you'll find significant theorems on polynomials and rational functions, rings of linear transformations on vector spaces (matrices) and, perhaps, some nonlinear analogs in commutative algebra (e.g. via Gröbner bases). Combinatorics makes heavy use of generating functions (formal power series). Algebraicized analysis employs differential/difference algebra to model elementary functions and their asymptotics (transseries), e.g. Liouville-Risch theory of integration/summation in finite terms, Rosenlicht's Hardy fields. Differential Galois theory studies solutions of differential equations and similarly for the difference analog (recurrences). Entire books have been written on rings of continuous functions (e.g. Gillman and Jerison). Abe Robinson's NSA (nonstandard analysis) is another algebraic approach to analysis.

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The question is not very well defined. For example, the nullstellensatz shows that the points in $\mathbb{C}^n$ (or any algebraically closed field) are in correspondence with the maximal ideals in $\mathbb{C}[x_1,...,x_n]$.

Indeed this correspondence between geometry and algebra (notice that we can talk about generalized points: prime ideals in $\mathbb{C}[x_1,...,x_n]$) is what's in the "core" of modern algebra. It's at the very least the main reason that we look at rings (arguably the primary reason we look at fields is Galois Theory; and Group Theory has intrinsic value).

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Well,

I think the Wedderburn's theorem, Gauss Lemma, and Eisentein's Criteria are good ones.

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