Original sentence: But when m is prime there are only two rings, both based on the cyclic group of order m
Edited sentence: A sufficient understanding of finite division rings will yield a good understanding of finite simple rings, therefore.As with finite groups, this question has also a deep connection with the choice of m.For instance, the classification of finite simple groups was a major breakthrough in the theory of finite groups.However, the classification of finite simple rings is necessarily trivial since any right artinian simple ring is isomorphic to a full matrix ring over a division ring.If m is the square of a prime, for instance, there are precisely eleven rings having order m.On the other hand, some aspects of the theory are much easier than that in finite groups.On the other hand, there can be only two groups having order m; both of which are abelian.The number of rings with m elements, for m a natual number, is listed under A027623 in the On-Line Encyclopedia of Integer Sequences.Therefore, any finite simple ring is isomorphic to a full matrix ring over a finite division ring.Therefore, the theory of finite rings, is in some respects, more complex than the theory of finite groups.When m is prime there are only two rings of order m; the additive group of each being isomorphic to the cyclic group of order m