Hereditary ring: Difference between revisions
(New page: {{curing property}} ==Definition== A commutative unital ring is termed '''hereditary''' if it satisfies the following equivalent conditions: * Every ideal in it is a [[projectiv...) |
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* Every submodule of a [[projective module]] is [[projective module|projective]] | * Every submodule of a [[projective module]] is [[projective module|projective]] | ||
* Every quotient of an [[injective module]] is [[injective module|injective]] | * Every quotient of an [[injective module]] is [[injective module|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]] | |||
Revision as of 14:44, 6 May 2008
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