TITLE: What are rings good for?
QUESTION [6 upvotes]: It's not a homework question or something, but I was wondering: groups are useful in physics and can be applied to many symmetry holding problems, fields can be used to construct vector spaces over the field. But what use do general rings have? Is it to construct modules?
Or more concretely, can someone give me an example of application of a ring.
Thanks in advance.
EDIT: Of course I know the trivial examples like $\mathbb{R}$, but since this is also a field it has its applications as a vector space or as a ground field for a vector space. What about $\mathbb{Z}[i]$ for example?

REPLY [2 votes]: The other answers have pointed out many applications of rings themselves, particularly to number theory. But by considering modules, rings can also shed a lot of light on vector spaces and on groups! They are a fantastic example of how, by making something more general (a ring is a generalisation of a field), we can gain a lot.
Let $M$ be a finitely generated module over a ring $R$ which is a PID. The structure theorem states that

There are uniquely determined ideals of $R$ $$I_1 \supset I_2\supset\cdots\supset I_n$$such that $$M\cong \bigoplus_{i=1}^nR/I_i$$

Two applications of this theorem are:

Taking $R = \mathbb Z$, we obtain the structure theorem for finitely generated abelian groups
Taking $R= \mathbb C[X]$, we obtain a neat and streamlined proof of Jordan normal form for complex matrices.