SCr modules over local rings

We show that the following two conditions, for each integer r greater than or equal to 1, are equivalent for a finitely generated module M over a complete Noetherian local ring (R, m): (a) For each integer q greater than or equal to 1, there is a submodule N-q subset of m(q)M such that M/N-q is embeddable in E-r, where E denotes the injective hull of the residue field R/m. (b) Either M subset of E-r, or else dim(R/m)Hom(R)(R/m, M)=k <r and there is no prime ideal P such that dimR/p = 1 and (R/p)(r-k+1) is embeddable in M. This is an extension of a result of M. Hochster (1977, Trans. Amer. Math. Sec. 231, 463-488), which is the case r = 1. We show other results related to these conditions. (C) 2001 Academic Press.