minimal primes of an ideal

using { t minimalPrimes}

To obtain a list of the minimal associated primes for an ideal I (i.e. the smallest primes containing I), use the function minimalPrimes.

i1 : R = QQ[w,x,y,z];

i2 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2)

               2           6    3                4         2    2
o2 = ideal (w*x  - 42y*z, x  + x z + 12w*y, - 47x z - 47x*z  + w )

o2 : Ideal of R

i3 : minimalPrimes I

                                                      3
o3 = {ideal (z, x, w), ideal (y, x, w), ideal (y, w, x  + z)}

o3 : List

If the ideal given is a prime ideal then minimalPrimes will return the ideal given.

i4 : R = ZZ/101[w..z];

i5 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);

o5 : Ideal of R

i6 : minimalPrimes I

                2           2   2          2      3   2   2      2 3      4 
o6 = {ideal (w*x  - 42y*z, x y*z  - 12w*x*z  + 11w , w x*z  + 47y z  - 43w ,
     ------------------------------------------------------------------------
      4       2      2   6    3
     x z + x*z  - 43w , x  + x z + 12w*y)}

o6 : List

See associated primes for information on finding associated prime ideals and primary decomposition for more information about finding the full primary decomposition of an ideal.