Deep decompositions of modules

A decomposition of an R-module into submodules, M = circle plus M-alpha, is called deep if for every submodule H of M, we have H = circle plus(H boole an AND M-alpha). We characterize when deep decompositions exist. We then sh ow that M similar or equal to circle plus M-P (over all maximal ideals P) i f and only if R/(Ann m) is a finite direct sum of quasi-local rings for all 0 not equal m is an element of M. We also show this decomposition is deep.