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: $Lm=0$.

Given two filters $F,G$ in the lattice $I_{l}R$ of left ideals on $R$, one defines their Gabriel composition (or Gabriel product) $F•G$ as the set of all left $R$-ideals $L$ such that there is $K$ in $G$ such that $K/L$ is $F$-torsion. 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.