A higher syzygy of a module with positive codimension is a maximal Cohenx2013;Macaulay module that plays an important role in Cohenx2013;Macaulay approximation over Gorenstein rings. We show that every maximal Cohenx2013;Macaulay module is a higher syzygy of some positive codimensional module if and only if the ring is an integral domain. Also we discuss the hierarchy of rings with respect to Cohenx2013;Macaulay approximation by codimensions of modules.