A general notion of algebraic entropy and the rank-entropy

We give a general definition of a subadditive invariant i of Mod(R), where R is any ring, and the related notion of algebraic entropy of endomorphisms of R-modules, with respect to i. We examine the properties of the various entropies that arise in different circumstances. Then we focus on the rank-entropy, namely the entropy arising from the invariant ‘rank’ for Abelian groups. We show that the rank-entropy satisfies the Addition Theorem. We also provide a uniqueness theorem for the rank-entropy.