There's a collection of results which most will learn in their first course on non-commutative rings which I recall being distinctly disappointed by and then heard that I was far from alone in this opinion.

When it comes to studying non-commutative rings it initially seems theoretically very interesting! When you work over fields, modules (in this case, vector spaces) have a very simple and lovely theory based around bases but lots of results from that context (even the existence of a basis) can fail when working over general non-commutative rings and it can be great fun to try and work out different theoretical requirements you could put on your ring to try and fix this (e.g. being a division ring, or semisimple or being an algebra over a field etc.).

However to actually see this difference in any meaningful way, you want a good supply of genuinely non-commutative rings which are somewhat reasonable in their properties:

• One example is that you might look at finite rings and we know that there is exactly one finite field of size equal to each prime power for each prime, so why not relax that condition to being just an integral domain but still commutative? It's not very difficult to argue that the finite integral domains are all fields...
• OK, so let's instead say that the rings we look at are division rings (i.e. every nonzero element has a right and left inverse but they aren't necessarily the same and the ring isn't necessarily the same). Unfortunately, a little more work gets you Wedderburn's Little Theorem which tells you that the finite division rings are all fields.
• OK let's even relax associativity! We'll look at finite alternative division rings, i.e. ones where $x(xy) = (xx)y$ and $y(xx) = (yx)x$ but all $x,y$ in the ring but $a(bc)$ doesn't have to equal $(ab)c$ for all $a,b,c$ in the ring. Well the Artin-Zorn Theorem tells you that in this case they are still fields.
• Another place you might look is real division algebras (which we suppose are associative but not commutative). $\mathbb{R}$ and $\mathbb{C}$ are rather simple examples, so you might try and apply the same kind of construction you performed on $\mathbb{R}$ to get $\mathbb{C}$ on $\mathbb{C}$ instead (this is the Cayley-Dickson construction). This works and gives you the quaternions which are very interesting and can help with understanding all kinds of things in group theory, number theory and physics.
• Let's try that again! Well actually you only get the octonions, which are still very interesting but they aren't actually distributive so don't really classify as what we're looking for. OK, are there any other finite-dimensional real division algebras? Frobenius' Theorem says that $\mathbb{R}, \mathbb{C}, \mathbb{H}$ are the only examples unfortunately.
• How about allowing potentially countably-infinite dimensional division algebras? Unfortunately this still only gives $\mathbb{R}, \mathbb{C},\mathbb{H}$ due to application of a result which has a number of names (in books I've read it goes variously by the non-commutative nullstellensatz or the Amitsur-Schur Lemma etc.).
• There are lots of similarly restrictive requirements on real division algebras even if we loosen the definition further. Suppose we allow them to only be alternative? Zorn proved you only get $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$.
• What if we allow it to not even be alternative but it must be a normed vector space with a norm which satisfies $\| ab\| = \|a\| \|b \|$? Hurwitz tells us you only get $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$.
• What if we require it to be associative and have a norm which satisfies $\|ab\| \leq \|a\| \|b\|$ and be complete (a Banach algebra)? Gelfand-Mazur tells you it's just $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$.
• The most general possible situation in this area is any real division algebra which isn't necessarily associative or alternative and no other requirements. Then Kervaire-Milnor tells you (by applying some deep results from algebraic topology) that they can still only be of dimensions 1,2,4 or 8 (and I think people feel there are more stringent requirements on such rings but this is still an area of research).

The outcome is that you struggle to prove things about these much more complex non-commutative rings but the moment you want to find some kind of family of them which is vaguely comprehensible (as opposed to, say, general matrix rings which are tough for a first course) you find none! I and a number of others felt that Ring Theory might be a lot richer if there had been finite, non-field division rings for example...