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