More results on congruent modules

Abstract

a b s t r a c t 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.

Jónsson and HS Modules over Commutative Rings

“…Lemma 55 (see [44,Theorem 2] Proof. Let > 0 be an ordinal, and assume that ℵ is not the successor of a cardinal of countable cofinality.…”

On elementarily κ-homogeneous unary structures

“…Many results were obtained in [8] and [7], including a complete characterization of the torsion-free congruent modules as well as some statements which were shown to be independent of ZFC. Most recently, the author studied this concept within the context of commutative algebra.…”