What are the major theorems about rings? The major theorems I've encountered so far require that a ring be at least an integral domain. What are the major theorems that flow from the ring axioms alone, and thereby motivate them?
 A: Take a look at this page about the landmark book Structure of Rings by Nathan Jacobson.
A: There is a handful (or maybe a few handfuls) of things like the isomorphism theorems, the Chinese Remainder theorem, and Nakayama's lemma that hold for all rings
But typically, you add (at least) one nice property to talk about interesting results. For example, you mentioned that "the results I've seen require the ring to be a domain".
Lots of famous theorems say something about the structure of the ring, or about the ring's modules. You will find many of these results in the Jacobson book that Andreas Caranti recommended.
I can only guess what theorems about domains you are thinking of, but some examples that do jump to mind are: principal ideal domains (PIDs) are Dedekind domains, which are in turn Noetherian domains. (These are types of ring structure results: they discuss conditions on elements and ideals within the ring.) An example about the module structure would be the fundemental theorem on finitely generated modules over a PID. 
There is an especially simple structure result on what commutative Artinian rings look like: they are all finite products of commutative local Artinian rings.
I think I would also consider the structure of finite fields to be a theorem about rings that everybody should know: for each prime $p$ and positive natural number $n$, there is exactly one field (up to isomorphism) of order $p^n$, and these are the only finite fields.
One result (which could be considered a module result) is the Artin-Wedderburn structure theorem which characterizes a certain property. Fields have the property that in all of their modules (=vector spaces), every submodule (=subspace) has a direct complement. You then ask "well, which rings have this property for their modules? We know fields do... are there more?". The answer turns out to be "The rings that do this are the ones characterized by the A-W theorem. The Jacobson density theorem is closely connected with the A-W theorem, and characterizes a property of a ring being right (or left) primitive.
The reach of "module" type theorems is illuminated by Morita's theorems which explain how equivalences of module categories translates into a relationship between Morita equivalent rings. In a nutshell, the theorems say that "R and S have equivalent module categories if R is isomorphic to $eM_n(S)e$ where $M_n(S)$ is a matrix ring over $S$ and $e$ is a "full idempotent" in that matrix ring.
Some very important (but rather unfamiliar to me) results exist for commutative local rings and special types of fields. These are intimately connected with the study of varieties in algebraic geometry, as they contain information about the geometry of surfaces. I think I have to leave it to someone more familiar with the theorems to cherry pick the representative ones.
A: My knowledge about rings is limited, but I find the following one particularly important:

$\mathbb{Z}$ is initial in the category $\text{Ring}$.

