Adequate domain

Symbol-free definition
An integral domain is termed an adequate domain if it is also an adequate ring.

Definition with symbols
An integral domain $$R$$ is termed an adequate domain if it satisfies the following conditions:


 * It is a Bezout domain
 * For any $$a,b \in R$$ with $$a \ne 0$$, there exist $$r,s \in R$$ such that $$a = rs$$, $$rR + bR = R$$ and if $$s'$$ is a non-unit (i.e. proper) divisor of $$s$$ then $$rR + s'R \ne R$$

Stronger properties

 * Principal ideal domain