-- Example 8.18, 8.27, 8.31
k = QQ
R = k[A..F]
M = genericSymmetricMatrix(R,3)
P = minors(2,M)
isPrime P -- it is the vanishing ideal of the image of a map A^3->A^6 (see p.121)
m = ideal vars R
isSubset(det M * m, P^2) -- in assumption that P^2 is primary, this implies that m^p \subset P^2 for some p and leads to a contradiction
Q1 = P^2 : A -- the primary component of the minimal associated prime P
Q1 == saturate(P^2,A)
Q1 == P^2 + det M 
P^2 : det M -- m is an associated prime... 
Q2 = m^4 -- ... m^4 is the corresponding primary component
P^2 == intersect(Q1,Q2)

-- blackbox M2 routines
decompose P^2 -- minimal associated primes
ass P^2
dec = primaryDecomposition P^2
first dec == Q1
last dec != m^4 -- note: Q2 in _not_ uniquely determined