Saturated subset: Difference between revisions
(New page: ==Definition== A subset in a commutative unital ring is termed a '''saturated subset''' if it satisfies the following equivalent conditions: * It contains <math>1</math>, it is [[mul...) |
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A subset in a [[commutative unital ring]] is termed a '''saturated subset''' if it satisfies the following equivalent conditions: | A subset in a [[commutative unital ring]] is termed a '''saturated subset''' if it satisfies the following equivalent conditions: | ||
* It contains <math>1</math>, it is [[multiplicatively closed subset|multiplicatively closed]], and if a product of two elements lies in the subset, so do both the elements. | * It contains <math>1</math> and does not contain <math>0</math>, it is [[multiplicatively closed subset|multiplicatively closed]], and if a product of two elements lies in the subset, so do both the elements. | ||
* It is the complement of a union of [[prime ideal]]s | * It is the complement of a union of [[prime ideal]]s | ||
Revision as of 22:23, 2 February 2008
Definition
A subset in a commutative unital ring is termed a saturated subset if it satisfies the following equivalent conditions:
- It contains and does not contain , it is multiplicatively closed, and if a product of two elements lies in the subset, so do both the elements.
- It is the complement of a union of prime ideals
