# saturate -- saturation of ideal or submodule

## Synopsis

• Usage:
saturate(I,J)
saturate I
• Inputs:
• I, an ideal, , or
• J, an ideal or . If not present, then J is the ideal generated by the variables of the ring
• Optional inputs:
• Outputs:
• an ideal, , or , the saturation of I with respect to J

## Description

If I is either an ideal or a submodule of a module M, the saturation (I : J^*) is defined to be the set of elements f in the ring (first case) or in M (second case) such that J^N * f is contained in I, for some N large enough.

For example, one way to homogenize an ideal is to homogenize the generators and then saturate with respect to the homogenizing variable.

 `i1 : R = ZZ/32003[a..d];` ```i2 : I = ideal(a^3-b, a^4-c) 3 4 o2 = ideal (a - b, a - c) o2 : Ideal of R``` ```i3 : Ih = homogenize(I,d) 2 2 3 2 3 2 o3 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d ) o3 : Ideal of R``` ```i4 : saturate(Ih,d) 2 2 3 2 3 2 o4 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d ) o4 : Ideal of R```
We can use this command to remove graded submodules of finite length.
 ```i5 : m = ideal vars R o5 = ideal (a, b, c, d) o5 : Ideal of R``` ```i6 : M = R^1 / (a * m^2) o6 = cokernel | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 | 1 o6 : R-module, quotient of R``` ```i7 : M / saturate 0_M o7 = cokernel | a a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 | 1 o7 : R-module, quotient of R```

If I and J are both monomial ideals, then a faster algorithm is used. If I or J is not a monomial ideal, generally Gröbner bases will be used to the compute the saturation. These will be computed as needed.

The computation is currently not stored anywhere: this means that the computation cannot be continued after an interrupt. This will be changed in a later version.