Free ideal ring

Definition for commutative rings

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

Relation with other properties

Stronger properties

• Field
• Principal ideal domain

Weaker properties

• Integral domain