Rings for which certain modules are CS Original Research Article

If image is any class of modules over a general ring R such that image is closed under direct sums, quotients and submodules, then every module in image is CS if and only if every module M in image has a decomposition M = circled plusiεI Mi, where each module Mi (i ε I) is either simple, or has length 2 and is X-injective or each module X in C. In consequence, necessary and sufficient conditions are given for a ring to have all its right singular modulesCS. Rings whose finitely generated modules are CS are also studied.