Free ideal ring

Symbol-free definition
A commutative unital ring (or any commutative ring) is termed a free ideal ring or fir if every ideal of that ring is a free module as a module over that ring.

Definition for noncommutative rings
For noncommutative rings, there are two notions:


 * Free left ideal ring or left fir
 * Free right ideal ring or right fir

Stronger properties

 * Field
 * Principal ideal domain

Weaker properties

 * Integral domain