# 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.

Title multiplicative filter MultiplicativeFilter 2013-03-22 16:48:22 2013-03-22 16:48:22 jocaps (12118) jocaps (12118) 6 jocaps (12118) Example msc 03E99 msc 54A99 Gabriel Filter Multiplicative Filter