Apparatus and methods are provided for use with a prosthetic valve that is designated for implantation at a patient's native heart valve, including a valve ring having a plurality of ring segments, each of the segments being hingedly coupled to an adjacent segment at a pivot joint. The valve ring is placed adjacent to a surface of the native heart valve, the prosthetic valve having been coupled to the valve ring. In an expanded state thereof, the valve ring defines a ring, all of the pivot joints being disposed in a plane that is perpendicular to a longitudinal axis of the ring. The valve ring is foldable into a shape that has a generally circular cross-section that defines and surrounds at least in part a central lumen, by folding the segments with respect to each other, at the pivot joints. Other embodiments are also described.