Here is a small Macaulay2 session</a> where I calculated the primary decomposition of an ideal. 
I used a session like this as I prepared the lecture on primary decomposition. 
All of the commands I used are explained on-line.


Find a computer on which Macaulay2 has been installed.
The command to start Macaulay2 is M2.

kustin@kustin-Latitude-E6520:~$ M2
Macaulay2, version 1.4
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : R=QQ[b,c,d,e]

o1 = R

o1 : PolynomialRing

i2 : M=matrix{{0,b,d},{b,c,e}}

o2 = | 0 b d |
     | b c e |

             2       3
o2 : Matrix R  <--- R

i3 : I=minors(2,M)

              2
o3 = ideal (-b , -b*d, - c*d + b*e)

o3 : Ideal of R

i4 : associatedPrimes(I)

o4 = {ideal (c, b), ideal (d, b)}

o4 : List

i6 : I:ideal(b)

o6 = ideal (d, b)

o6 : Ideal of R

i7 : I:ideal(d^2)

o7 = ideal (c, b)

o7 : Ideal of R

i8 : intersect(ideal (I,d^2),ideal(b,c))

                             2
o8 = ideal (c*d - b*e, b*d, b )

o8 : Ideal of R

i9 : o8==I

o9 = true

i10 : primaryDecomposition(I)

                             2                   2
o10 = {ideal (c, b), ideal (d , c*d - b*e, b*d, b )}

o10 : List

i11 :