Integral domain in which every irreducible is prime

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

An integral domain in which every irreducible is prime is an integral domain satisfying the following equivalent condition: every irreducible element in it is a prime element.

Relation with other properties

Stronger properties

• Euclidean domain
• Principal ideal domain
• Bezout domain
• gcd domain
• Unique factorization domain

Weaker properties

• Integral domain in which any two irreducible factorizations are equal upto ordering and associates