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
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i3 : minimalPrimes I
3
o3 = {ideal (z, x, w), ideal (y, x, w), ideal (y, w, x + z)}
o3 : List
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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
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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.