Algebraic entropy of endomorphisms over local one-dimensional domains

Let R be a local one-dimensional integral domain, with maximal ideal and field of fractions Q. Here, a local ring is not necessarily Noetherian. We consider the algebraic entropy entg, defined using the invariant gen, where, for M a finitely generated R-module, gen(M) is its minimal number of generators. We relate some natural properties of R with the algebraic entropies entg(φ) of the elements φ ε Q, regarded as endomorphisms in EndR(Q). Specifically, let R be dominated by an Archimedean valuation domain V, with maximal ideal P. We examine the uniqueness of V, the transcendency of the residue field extension V/P over R/ℳ, and the condition for R to be a pseudo-valuation domain. We get mutual information between these properties and the behavior of entg, focusing on the conditions entg(φ) = 0 for every φ ε Q, ent g(ψ) = ∞ for some ψ ε Q, and ent g(φ) < ∞ for every φ ε Q. © 2009 World Scientific Publishing Company.