Wedderburn.27s theorems Finite ring

wedderburn s little theorem asserts finite division ring commutative:

if every nonzero element r of finite ring r has multiplicative inverse, r commutative (and therefore finite field).

nathan jacobson later discovered yet condition guarantees commutativity of ring: if every element r of r there exists integer n > 1 such r  = r, r commutative. more general conditions guarantee commutativity of ring known.

yet theorem wedderburn has, consequence, result demonstrating theory of finite simple rings relatively straightforward in nature. more specifically, finite simple ring isomorphic ring




m

n


(


f


q


)


{\displaystyle m_{n}(\mathbb {f} _{q})}

of n n matrices on finite field of order q. follows 2 theorems of joseph wedderburn established in 1905 , 1907 (one of wedderburn s little theorem). on other hand, classification of finite simple groups 1 of major breakthroughs of twentieth century mathematics, proof spanning thousands of journal pages. therefore, in respects, theory of finite rings simpler of finite groups.