minimal primes of an ideal

using 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 (x  + z, w, y), ideal (x, w, y), ideal (x, z, w)}

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           4       2      2   2   2          2      3 
o6 = {ideal (w*x  - 42y*z, x z + x*z  - 43w , x y*z  - 12w*x*z  + 11w ,
     ------------------------------------------------------------------------
        2   2    2 3      4   6    3
     43w x*z  + y z  - 31w , x  + x z + 12w*y)}

o6 : List

warning

Warning (15 May 2001): If you stop a function mid process and then run minimalPrimes an error is given. Restarting Macaulay 2 and then running minimalPrimes works around this.

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