Use intersect to compute the intersection of two or more ideals.
i1 : R = QQ[a..d];
i2 : intersect(ideal(a,b),ideal(c*d,a*b),ideal(b*d,a*c))
o2 = ideal (b*c*d, a*c*d, a*b*d, a*b*c)
o2 : Ideal of R
The command intersect will only work with proper ideals. To intersect an ideal with a ring, use selectInSubring along with the elimination ordering, see Eliminate.