nLab Gabriel composition of filters

Given a filter $F$ of ideals in a ring $R$, a left $R$-module $M$ is $F$-torsion (Gabriel‘s terminology: $F$-negligible) if for any $m\in M$ there exists a $L$ in $F$ which annihilates it: $L m=0$.

Given two filters $F,G$ of left ideals on $R$, one defines their Gabriel composition (or Gabriel product) $F\bullet G$ as the set of all left ideals $L$ in $R$ such that there is $K$ in $G$ with $K/L$ being $F$-torsion. Then $F\bullet G$ is again a filter of left ideals in $R$.

Gabriel composition of filters corresponds to the Gabriel multiplication of the corresponding torsion classes considered as strictly full subcategories. A Gabriel composition of uniform filters is uniform.