-- This program calculates the product and intersection of three ideals I, J, K.
-- In the ring R,the polynomial ring in three variables over the finite field
-- of 31991 elements.
-- It verifies that the radical of the product is the intersection. It varifies that
-- the intersection is I and hence we 'probably' got the components of Z(I).
-- In fact, Macaulay2 (that you run from prism by writing the commant M2),
-- can find the components directly. Use the command 'decompose I' for that.
-- To execute this file you use    input "filename"
-- The should be placed in the appropriate directory. Experiment!

R=ZZ/31991[x, y, z];
I = ideal(x*z - x, x^2-y*z);
J= ideal(x, y);
K=ideal(x, z);
L =ideal(z - 1, x^2 - y*z);
M1=J*K*L
M2=radical(M1)
M1==M2
M3=intersect(J, K, L)
M2==M3
decompose I
t = x^3 - y*z;
t % I
s = x^2*(x*z - x) + z^3*(x^2 - y*z);
s%I