More results on congruent modules

W.R. Scott characterized the infinite abelian groups G for which H≅G for every subgroup H of G of the same cardinality as G W.R. Scott, On infinite groups, Pacific J. Math. 5 (1955) 589-598]. In G. Oman, On infinite modules M over a Dedekind domain for which N≅M for every submodule N of cardinality |M|, Rocky Mount. J. Math. 39 (1) (2009) 259-270], the author extends Scott’s result to infinite modules over a Dedekind domain, calling such modules congruent, and in a subsequent paper G. Oman, On modules M for which N≅M for every submodule N of size |M|, J. Commutative Algebra (in press)] the author obtains results on congruent modules over more general classes of rings. In this paper, we continue our study.