RINGS FOR WHICH CERTAIN MODULES ARE CS

If C is any class of modules over a general ring R such that C is clos ed under direct sums, quotients and submodules, then every module in C is CS if and only if every module M in C has a decomposition M = (+i epsilon l) M(i), where each module M(i) (i epsilon I) is either simple , or has length 2 and is X-injective for each module X in C. In conseq uence, necessary and sufficient conditions are given for a ring to hav e all its right singular modules CS. Rings whose finitely generated mo dules are CS are also studied.