Hereditary ring

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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

A commutative unital ring is termed hereditary if it satisfies the following equivalent conditions:

• Every ideal in it is a projective module
• Every submodule of a free module is projective
• Every submodule of a projective module is projective
• Every quotient of an injective module is injective
• The global dimension of the ring is at most one

Relation with other properties

Stronger properties

• Field
• Dedekind domain: In fact, an integral domain is hereditary if and only if it is a Dedekind domain
• Semisimple Artinian ring: This is a ring with global dimension zero

Weaker properties

• Semihereditary ring