Simply Presented Valuated Modules

✍ Scribed by C.R. Merrin

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 960 KB
• Volume: 168
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1994.1245

✦ Synopsis

Let (R) be a principal ideal domain and let (p) be a fixed prime in (R). We show how from a valuated forest (F) a (p)-valuated (R)-module (S(F)) may be derived, and then we discuss the basic properties of (S(F)). We develop and explore the concept of levels in (R)-modules, and this investigation leads to some useful observations about endomorphisms of (R)-modules. Two principal results are that a valuated torsion-free tree (T) is irretractable if and only if (S(T)) is indecomposable as a (p)-valuated (R)-module, and if (F) and (F^{\prime}) are forests consisting of reduced irretractable valuated torsion-free trees and (S(F) \cong S\left(F^{\prime}\right)), then (F \cong F^{\prime} .1994) Academic Press. Inc.