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

i1 : R = QQ[a..d]; |

i2 : I = ideal(a^2*b-c^2, a*b^2-d^3, c^5-d); o2 : Ideal of R |

i3 : J = ideal(a^2,b^2,c^2,d^2); o3 : Ideal of R |

i4 : I:J 2 3 2 2 5 o4 = ideal (a*b - d , a b - c , c - d) o4 : Ideal of R |

i5 : P = quotient(I,J) 2 3 2 2 5 o5 = ideal (a*b - d , a b - c , c - d) o5 : Ideal of R |

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.

i6 : R = ZZ/32003[a..d]; |

i7 : I = ideal(a^3-b, a^4-c) 3 4 o7 = ideal (a - b, a - c) o7 : Ideal of R |

i8 : Ih = homogenize(I, d) 2 2 3 2 3 2 o8 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d ) o8 : Ideal of R |

i9 : saturate(Ih, d) 2 2 3 2 3 2 o9 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d ) o9 : Ideal of R |