It had long puzzled me why people usually use the ideal product to show that the intersection of any two Zariski-open subsets of the prime spectrum of a ring is again a Zariski-open subset – after all, the intersection of the two ideals would do just as well. But perhaps this is the reason:
Proposition. Let f : A → B be a ring homomorphism and let I and J be ideals of A. Then the ideal of B generated by f(IJ) is the ideal product of the ideals generated by fI and fJ.
It is well known that the poset of ideals of a ring has arbitrary joins (namely, the ideal sum) and that they are preserved by pushforward along ring homomorphisms. Moreover, the ideal product distributes over the ideal sum, so the poset of ideals of a ring in fact a quantale, and we have a functor CRing → Quant. A result of Mulvey says that the frame of Zariski-open subsets of the prime spectrum of A is the frame reflection of the quantale of ideals of A.
While it is true that the poset of ideals is a complete lattice, not all binary meets (i.e. intersections) are preserved by pushforward. (Consider the homomorphism k[x, y] / (xy) → k[z] / (z2) sending x and y to z.) Distributivity also fails. (Consider the ring k[x, y] / (x2, xy, y).) Still, one might ask, what is the relationship between the frame of Zariski-open subsets of the prime spectrum and the frame reflection of the poset of ideals, considered as a join semilattice with finite meets?