Let $E$ be a principal ideal domain and $A$ an $E$-algebra which is finitely generated as a module. From the structure theorem we are able to classify the module structure of $A$ up to isomorphism, but I wonder if we can do better, that is, to classify also the multiplicative structure as well.
I came up with this question when I am trying to characterize rings of the form $\mathbb Z^m\times \mathbb Z/p_1^{k_1}\mathbb Z\times \mathbb Z/p_2^{k_2}\mathbb Z\times\cdots\times \mathbb Z/p_n^{k_n}\mathbb Z$ (where we assume it is possible to have $p_i=p_j$). This is a special case of the previous paragraph: Can we classify finite algebras over the ring $\mathbb Z$? What’s the best we can get?
As I was typing down this question I realize this might be very difficult. Finite commutative rings are finite $\mathbb Z$-algebras, but there is no effective classification till now.
Finite algebras over integers = finite associative rings. There cannot be any classification of these rings. In particular directly indecomposable finite associative rings and even subdirectly indecomposable ones are undescribable.
One can characterise semisimple finite associative rings as direct products of rings of matrices over finite fields. But associative finite nil (=nilpotent) rings are as complicated as finite Lie nilpotent rings or finite nilpotent groups.