Finitely generated ideal
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
This property of ideals in commutative unital rings depends only on the ideal, viewed abstractly as a module over the commutative unital ring. The corresponding module property that the ideal must satisfy is: finitely generated module
View other such properties
An ideal in a commutative unital ring is said to be finitely generated if it has a finite generating set, that is, if there is a finite set such that it is the smallest ideal containing that finite set.
Definition with symbols
Relation with other properties
This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property
In general, an intersection of finitely generated ideals need not be finitely generated. However, for Noetherian rings, where every ideal is finitely generated, an intersection of finitely generated ideals is certainly finitely generated.