Adequate domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

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 $s^{\prime }$ is a non-unit (i.e. proper) divisor of $s$ then $rR+s^{\prime }R\ne R$

Relation with other properties

Stronger properties

• Principal ideal domain