In 2002, Ilan Hirshberg defined a C*-algebra associated with endomorphisms of abelian groups with finite cokernel. In fact, it was a starting point of new constructions associating C*-algebras to simple objects like groups, semigroups and rings. In this book, we present a natural continuation of that work, we construct the same C*-algebra for endomorphisms with infinite cokernel. We use the subset structure of the group to index interesting projections in the C*-algebra and present interesting connections with the semigroup C*-algebras defined by Xin Li. We also calculate some K-groups associated to such constructions.