Finite Rings - Wedderburn's Theorems

Wedderburn's Theorems

There are other deep aspects to the theory of finite rings, apart from mere enumeration. For instance, Wedderburn's little theorem asserts that any finite division ring is necessarily commutative (and therefore a finite field). Nathan Jacobson later discovered yet another condition which guarantees commutativity of a ring:

If for every element r of R there exists an integer n > 1 such that rn = r, then R is commutative.

If, r2 = r for every r, the ring is called a Boolean ring. More general conditions which guarantee commutativity of a ring are also known.

Yet another theorem by Wedderburn has, as its consequence, a result demonstrating that the theory of finite simple rings is relatively straightforward in nature. More specifically, any finite simple ring is isomorphic to the ring of n by n matrices over a finite field of order q. This follows from two theorems of Joseph Wedderburn established in 1905 and 1907 (one of which is Wedderburn's little theorem). On the other hand, the classification of finite simple groups was one of the major breakthroughs of twentieth century mathematics, its proof spanning thousands of journal pages. Therefore, in some respects, the theory of finite rings is simpler than that of finite groups.