In a previous thread, I asked a question different from that I actually intended to ask. Since this question is licit and was answered by micromass, I open this new thread. The right question is in fact: If R is an integral domain, and R' is INTEGRAL over R, then the function f which assigns to an ideal I' of R' the ideal I = I' ∩ R, sends surjectively the prime ideals of R' to the prime ideals of R, the maximal ideals of R' to the maximal ideals of R, and (not nessarily surjectively) a non prime ideal of R' to a non-prime ideal of R. [NOTE: THIS LAST CLAIM WAS PROVED TO BE FALSE IN THIS THREAD] It would be nice if it could be proved that it sends SURJECTIVELY a non-prime ideal to a non-prime ideal, or equivalently, if it could be proved that f is a surjection from the set of ideals of R' to the set of ideals of R. But for the moment, I can only see that every ideal I of R is included in a maximal ideal of R'; the example of micromass in the previous thread does not fit here since Z_(2) is not integral over Z. Any ideas for a proof or a counter example ? N.B: It is 2:00 in my country, so, I will react to possible rapid answers in several hours.