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 $ℐ(A)$be the set of all ideals of $A$.

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

• A Gabriel Filter of $A$ is a filter $ℱ$ on $ℐ(A)$such that

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

Note that Gabriel Filters are also Multiplicative Filters.