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.