The quotient of two ideals $I, J\subset R$ is ideal $I:J$ of elements $f\in R$ such that $f J \subset I$.





The functions : and quotient perform the same basic operation, however quotient takes options.
The saturation of an ideal $I\subset R$ with respect to another ideal $J\subset R$ is the ideal $I:J^\infty$ of elements $f\in R$ such that $f J^N\subset I$ for some $N$ large enough. If the ideal $J$ is not given, the ideal generated by the variables of the ring $R$ is used.
For example, one way to homogenize an ideal is to homogenize the generators and then saturate with respect to the homogenizing variable.



