Abstract/formal interest of rings

Question

I am about to introduce first year undergrads to the concept of rings, after spending some time looking at groups; and I would like to give them more than a practical motivation (the most usual rings they have been manipulating: real, complex, rational numbers and matrices) so also a more abstract motivation: why is it formally interesting to extend groups to rings, to add a ``denser'' structure?

I have a physics background and I definitely lack examples or intuition to answer that. What can you think of?

Answer 1

This sort of thing is usually best done by example, rather than from the top down.

The basic goal is to convince them that rings are a very good level of generalization of integers and real numbers. Those things are very useful, and you want to say that the generalizations can be useful too.

Here are the applications of rings that I find very compelling:

1. You already mentioned matrices, I know, but I don't want you to write them off. If you can establish the connection of matrices to the ring of linear transformations of a vector space, you have already done a lot. I'm not sure anybody can fully appreciate rings until they realize what linear algebra can do.

2. polynomial rings: these are essential for doing algebraic geometry and field theory. You might be able to single out a classical result of field theory and steer toward it through polynomial rings.

3. Representation theory: if you think groups are already motivated, you could try to also motivate how group representations are interesting. (This is especially connected to physics.) Then you can make a little miracle happen: $F$ representations of $G$ naturally become modules of the group ring $F[G]$. Suddenly, linear algebra can be applied to group representations.

4. Cyclic codes: if you have time to motivate linear codes, the cyclic linear codes are in correspondence with the ideals of $F[x]/(x^n)$. Knowing the ring theory behind this tells you exactly how to build all possible linear codes of length $n$, as well as some properties the code is going to have.

Answer 2

The nLab describes rings like this:

More abstractly, a ring is a monoid internal to abelian groups (with their tensor product of abelian groups), and this perspective helps to explain the central relevance of the concept, owing to the fundamental nature of the notion of monoid objects.

You won't want to use these words, but you could say something like this: we know addition (abelian groups) is important; multiplication is also important (as when we compose two functions to get another function); multiplication in the context of addition leads naturally to the definition of a ring.