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