There are two notions associated to the colon construction for modules.
Similar to the case of ideals, the quotient of two $R$modules $M, N$ contained in the same ambient module is an ideal $M:N$ of elements $f\in R$ such that $f N \subset M$. This is equivalent to the annihilator of the quotient module $(M+N)/M$.
The quotient of an $R$module $M\subset F$ with respect to an ideal $J\subset R$ is the module $M:_F J$ of elements $f\in F$ such that $J f\subset M$.
The saturation of an $R$module $M\subset F$ with respect to an ideal $J\subset R$ is an $R$module $M:_F J^\infty$ of elements $f\in F$ such that $J^N f\subset M$ for some $N$ large enough. If the ideal $J$ is not given, the ideal generated by the variables of the ring $R$ is used.
If $M=M:_F J^\infty$ (or, equivalently, $M=M:_F J$), we say that $M$ is saturated with respect to $J$. We can use this command to remove graded submodules of finite length.




The annihilator of an $R$module $M$ is the ideal $\mathrm{ann}(M) = \{ f \in R  f M = 0 \}$.



You may also use the abbreviation ann:
