Corner’s Realization Theorems from the Viewpoint of Algebraic Entropy

Abstract

The realization theorems for reduced torsion-free rings as endomorphism rings of reduced torsion-free Abelian groups, proved by Corner in his celebrated papers, are applied to the rings of integral polynomials \(\mathbb{Z}[X]\) and the power series ring \(\mathbb{Z}[[X]]\), and are compared with another realization theorem proved in Corner’s paper on Hopficity in torsion-free groups, and with some variation of his results. The \(\mathbb{Z}[X]\)-module structure of the groups obtained from these different constructions is investigated looking at the cyclic trajectories of their endomorphisms, and at the corresponding values of the intrinsic algebraic entropy \(\widetilde{\mathrm{ent}}\).