Some finiteness chain conditions on the set of intermediate rings

Let R⊂S be an extension of integral domains and [R,S] be the set of intermediate rings between R and S. We say that [R,S] satisfies the finite chain condition (FCC) if every chain of distinct intermediate rings between R and S is finite. Our main purpose is to determine necessary and sufficient conditions so that [R,S] satisfies FCC. We also investigate the relationship between FCC and other finiteness conditions. Several satisfactory results are settled, but special attention is focused on the case where R is a Prüfer ring or a Noetherian ring.