Butler Modules Over Prüfer Domains

Abstract

If R is an integral domain, let  be the class of torsion free completely decomposable R-modules of finite rank. Denote by  the class of those torsion-free R-modules A such that A is a homomorphic image of some C ∊ , and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ∊ . Further, let Q  and Q 𝒫 be the respective closures of  and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q  = Q 𝒫, and  = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if 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  = Q  = Q 𝒫 = 𝒫 if and only if R is almost maximal.

Key Words:

• Prüfer domains
• Torsion-free modules of finite rank

2000 Mathematics Subject Classification:

• Primary 13C13
• Secondary 13F05

Notes

Communicated by A. Facchini.