Finitely generated ideal

Symbol-free definition
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
An ideal $$I$$ in a commutative unital ring $$R$$ is said to be finitely generated if there is a finite set $$x_1, x_2, ..., x_n$$ such that $$I = Rx_1 + Rx_2 + ... + Rx_n$$.

Stronger properties

 * Principal ideal
 * Noetherian ideal

Metaproperties
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.