Unique irredundant intersections of completely irreducible ideals

An ideal of a commutative ring is completely irreducible if it is not the intersection of any set of proper overideals. It is known that every ideal is an intersection of completely irreducible ideals. We characterize the rings for which every ideal can be represented uniquely as an irredundant intersection of completely irreducible ideals as precisely the rings in which every proper ideal is an irredundant intersection of powers of maximal ideals. We prove that every nonzero ideal of an integral domain R has a unique representation as an intersection of completely irreducible ideals if and only if R is an almost Dedekind domain with the property that for each proper ideal A the ring RA has at least one finitely generated maximal ideal. We characterize the rings for which every proper ideal is an irredundant intersection of powers of prime ideals as precisely the rings R for which (i) RM is a Noetherian valuation ring for each maximal ideal M, and (ii) every ideal of R is an irredundant intersection of irreducible ideals.