Algebraic entropy for Abelian groups

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In this paper the algebraic entropy, introduced in 1965 by Adler, Konheim and McAndrew, is studied. The so-called Addition Theorem is proved. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all whose endomorphisms have zero algebraic entropy. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is also proved.