minimal primes of an ideal

minimal primes of an ideal

Sections:

using decompose

warning

using decompose

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

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 : decompose I



                                                      3

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



o3 : List

If the ideal given is a prime ideal then decompose 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 : decompose I



                2           4       2      2   2   2          2      3     2   2    2 3      4   6    3

o6 = {ideal (w*x  - 42y*z, x z + x*z  - 43w , x y*z  - 12w*x*z  + 11w , 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 decompose an error is given. Restarting Macaulay 2 and then running decompose 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.