Corner’s Realization Theorems from the Viewpoint of Algebraic Entropy

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 ℤ [ X ] $$\mathbb{Z}[X]$$ and the power series ring ℤ [ [ X ] ] $$\mathbb{Z}[[X]]$$ , 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 ℤ [ X ] $$\mathbb{Z}[X]$$ -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 e n t ̃ $$\widetilde{\mathrm{ent}}$$ .