Structure theorem of finite rings

Solution 1:

There is no known complete classification of finite rings.

In the following, I'm assuming a ring is unitary, but not necessarily commutative.

The first step is to see that the decomposition of the additive group into groups of pairwise coprime prime power orders is also a ring-theoretic decomposition. In this way, we get a unique decomposition of any finite ring into rings of prime power order. Hence it suffices to classify rings of order $p^n$ with $p$ prime.

For $n=1$, there is only the ring $\mathbb Z/p\mathbb Z$.

For $n=2$, it's a nice exercise to show that up to isomorphism, there are the 4 rings $\mathbb Z/p^2\mathbb Z$, $\mathbb F_{p^2}$, $\mathbb Z/p\mathbb Z \times \mathbb Z/p\mathbb Z$ and $\mathbb Z/p\mathbb Z[X]/(X^2)$. See here and here.

The case $n = 3$ is already quite tedious. It can be found in Theorem 14 of the article R. Raghavendran, Finite Associative Rings, Composito Mathematica 21 (1969), 195—229. The result is that there are 11 rings of order 8 and 12 rings of order $p^3$ for $p$ an odd prime. The only non-commutative ring among them is the ring of upper $2\times 2$ triangular matrices over $\mathbb F_p$.

If I remember correctly, today the classification is known up to $p^5$ or $p^6$. It gets increasingly nasty, of course.

For some further classification-related properties of finite rings, see this discussion.