Butler Modules Over Prüfer Domains

Abstract

If R is an integral domain, let 𝒞 be the class of torsion free completely decomposable R-modules of finite rank. Denote by ℛ the class of those torsion-free R-modules A such that A is a homomorphic image of some C ∈ 𝒞, and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ∈ 𝒞. Further, let Q ℛ and Q 𝒫 be the respective closures of ℛ and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q ℛ = Q 𝒫, and ℛ = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ∈ 𝒞 has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then ℛ = Q ℛ = Q 𝒫 = 𝒫 if and only if R is almost maximal.