8.1 Finite rings

We recall some basic facts about finite commutative rings with identity (and 1 $\neq$ 0). (The adventurous reader may like to explore which parts of this entire section can be carried over to the non-commutative case). The reader can prove these results from first principles.

1. 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''.
2. Any prime ideal in a finite ring is maximal.
3. (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.
4. In a finite local ring every element is either a unit or nilpotent. Moreover, a finite local ring has pⁿ elements for some prime p and some integer n.
5. The residue field of a finite local ring is the quotient of the ring by its maximal ideal. This is a finite field.
6. An element of a finite local ring is a unit if and only if its image in the residue field is non-zero.