- In any finite ring there are finitely many ideals and in particular there are finitely many maximal ideals. In other words such a ring is ``semi-local''.
- Any prime ideal in a finite ring is maximal.
- (Analogue of Chinese Remainder Theorem). Any finite ring is a
product of finite
*local*rings; i. e. finite rings which have only one maximal ideal. - In a finite local ring every element is either a unit or
nilpotent. Moreover, a finite local ring has
*p*^{n}elements for some prime*p*and some integer*n*. - The
*residue field*of a finite local ring is the quotient of the ring by its maximal ideal. This is a finite field. - An element of a finite local ring is a unit if and only if its image in the residue field is non-zero.