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# 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MinimalPrimes/doc.m2:310:0.