PrimaryDecomposition -- primary decomposition and associated primes routines for ideals and modules

i1 : R = ZZ/32003[a..d];

i2 : I = monomialCurveIdeal(R,{1,3,4})

                        3      2     2    2    3    2
o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o2 : Ideal of R

i3 : J = ideal(a^3,b^3,c^3-d^3)

             3   3   3    3
o3 = ideal (a , b , c  - d )

o3 : Ideal of R

i4 : I = intersect(I,J)

             4    3      3    3      4      3         3      4   6      3 2  
o4 = ideal (b  - a d, a*b  - a c, b*c  - a*c d - b*c*d  + a*d , c  - b*c d  -
     ------------------------------------------------------------------------
      3 3      5     5    2 3       2 3    2 4   2 4    3 3    3 3    2   3 
     c d  + b*d , a*c  - b c d - a*c d  + b d , a c  - a d  + b d  - a c*d ,
     ------------------------------------------------------------------------
      3 3    3 3     2 3    3   2    3   2      2 3   2   3    3 2     3 2   
     b c  - a d , a*b c  - a c*d  + b c*d  - a*b d , a b*c  - a c d + b c d -
     ------------------------------------------------------------------------
      2   3   3 3    3   2   4 2    3 2
     a b*d , a c  - a b*d , a c  - a b d)

o4 : Ideal of R

i5 : removeLowestDimension I

                        3      2     2    2    3    2
o5 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o5 : Ideal of R

i6 : topComponents I

                        3      2     2    2    3    2
o6 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o6 : Ideal of R

i7 : radical I

                          2    2    3    2    6    3 3    2 4      5
o7 = ideal (b*c - a*d, a*c  - b d, b  - a c, c  - c d  - b d  + b*d )

o7 : Ideal of R

i8 : minimalPrimes I

                         3      2     2    2    3    2                    
o8 = {ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c), ideal (c - d, b,
     ------------------------------------------------------------------------
                       2          2
     a), ideal (b, a, c  + c*d + d )}

o8 : List

i9 : associatedPrimes I

                         3      2     2    2    3    2                    
o9 = {ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c), ideal (c - d, b,
     ------------------------------------------------------------------------
                       2          2
     a), ideal (b, a, c  + c*d + d )}

o9 : List

i10 : primaryDecomposition I

                          3      2     2    2    3    2                   3 
o10 = {ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c), ideal (c - d, b ,
      -----------------------------------------------------------------------
       3           2          2   3   3
      a ), ideal (c  + c*d + d , b , a )}

o10 : List