-- Here we demonstrate how Macaulay2 can implement many of the
-- operations described in Chater 4 of CLS

R = QQ[x,y,z, MonomialOrder=>Lex];

I = ideal( x*y^2, x*z );
J = ideal( x );
K = ideal( y, z );

-- Ideal sum:
I + J

-- Ideal product:
I * J

-- Ideal powers (i.e. iterated products I*I*...*I):
I^2

-- Ideal intersection:
intersect(I,J)

-- Ideal quotient:
quotient(I,K)

-- Ideal saturation:
saturate(I,K)

-- Radical of an ideal:
radical(I)

-- The primary decomposition of an ideal:
primaryDecomposition(I)

-- lcms and gcds of polynomials
lcm( x^2*y, x*y*z^3 )
gcd( x^2*y, x*y*z^3 )