SCr Modules over Local Rings

We show that the following two conditions, for each integer r ≥ 1, are equivalent for a finitely generated module M over a complete Noetherian local ring (R, ):• (a) For each integer ≥ 1, there is a submodule such that / is embeddable in , where denotes the injective hull of the residue field / .• (b) Either , or else (/ , ) = < and there is no prime ideal such that / = 1 and (/ ) is embeddable in .This is an extension of a result of M. Hochster (1977, Trans. Amer. Math. Soc.231, 463–488), which is the case r = 1. We show other results related to these conditions.