multiplicative filter
For any ring $A$, any set $S\subset A$ and any element $x\in A$, we use the notation
$$(S:x):=\{a\in Aax\in S\}$$ 
Let $A$ be a commutative ring with unity, and let $\mathcal{I}(A)$ be the set of all ideals of $A$.

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

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A Gabriel Filter of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that
$$[I\in \mathcal{F},J\in \mathcal{I}(A)\text{and}\forall x\in I,(J:x)\in \mathcal{F}]\Rightarrow J\in \mathcal{F}$$
Note that Gabriel Filters are also Multiplicative Filters.
Title  multiplicative filter 

Canonical name  MultiplicativeFilter 
Date of creation  20130322 16:48:22 
Last modified on  20130322 16:48:22 
Owner  jocaps (12118) 
Last modified by  jocaps (12118) 
Numerical id  6 
Author  jocaps (12118) 
Entry type  Example 
Classification  msc 03E99 
Classification  msc 54A99 
Defines  Gabriel Filter 
Defines  Multiplicative Filter 