# 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.