multiplicative filter

For any ring $A$ , any set $S\subset A$ and any element $x\in A$ , we use the notation

(S:x):=\{a\in A\ ax\in S\}

Let $A$ be a commutative ring with unity, and let $\mathcal{I}(A)$ be the set of all ideals of $A$ .

•

A Multiplicative Filter of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that $I,J\in\mathcal{F}\Rightarrow IJ\in\mathcal{F}$ .
•

A Gabriel Filter of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that

$[I\in\mathcal{F},J\in\mathcal{I}(A)\textrm{ and }\forall x\in I,(J:x)\in% \mathcal{F}]\Rightarrow J\in\mathcal{F}$

Note that Gabriel Filters are also Multiplicative Filters.