Finite Rings

Finite Rings

In mathematics, more specifically abstract algebra, a finite ring is a ring (not necessarily with a multiplicative identity) that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history.

As with finite groups, the complexity of the classification depends upon the complexity of the prime factorization of m. If m is the square of a prime, for instance, there are precisely eleven rings having order m. On the other hand, there can be only two groups having order m; both of which are abelian.

The theory of finite rings is more complex than that of finite abelian groups, since any finite abelian group is the additive group of at least two nonisomorphic finite rings: the direct product of copies of, and the zero ring. On the other hand, the theory of finite rings is simpler than that of not necessarily abelian finite groups. For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, any finite simple ring is isomorphic to the ring of n by n matrices over a finite field of order q.

The number of rings with m elements, for m a natural number, is listed under  A027623 in the On-Line Encyclopedia of Integer Sequences.