Simply presented valuated modules (Q1335087)

A forest is a set \(F\) together with a partial function \(\pi: F\to F\) such that for all \(n\geq 1\) and \(x\) in the domain of \(\pi\), \(\pi^ n x= x\to \pi x= x\). The author defines an equivalence relation \(\sim\): If \(x,y\in F\), \(x\sim y\) if there is \(z\in F\) such that there exist integers \(m\), \(n\), with \(x= \pi^ n z\) and \(y= \pi^ m z\). A forest with one equivalence class is a tree. Let Ord be the class of ordinals, a valuated forest \((F,\nu)\) is a forest \(F\) with a function \(\nu: F\to \text{Ord}\cup \{\infty\}\) with some properties. Let \(R\) be a principal ideal domain and let \(p\) be a fixed prime in \(R\). Then, the author defines a functor \(F\ldots \to S(F)\) from the category of forests to the category of \(p\)-valuated modules. There are two main results: A valuated torsion-free tree \(T\) is irretractable iff \(S(T)\) is indecomposable as a \(p\)-valuated \(R\)-module. If \(F\) and \(F'\) are forests consisting of reduced irretractable valuated torsion-free trees and \(S(F)\cong S(F')\), then \(F\cong F'\).