Simply Presented Valuated Modules Original Research Article

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′ are forests consisting of reduced irretractable valuated torsion-free trees and S(F) congruent with S(F′), then F congruent with; F′.