We introduce the algebraic entropy for endomorphisms of arbitrary abelian
groups, appropriately modifying existing notions of entropy. The basic
properties of the algebraic entropy are given, as well as various examples. The
main result of this paper is the Addition Theorem showing that the algebraic
entropy is additive in appropriate sense with respect to invariant subgroups.
We give several applications of the Addition Theorem, among them the Uniqueness
Theorem for the algebraic entropy in the category of all abelian groups and
their endomorphisms. Furthermore, we point out the delicate connection of the
algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory.