Corner’s Realization Theorems from the Viewpoint of Algebraic Entropy

Brendan Goldsmith () and Luigi Salce ()
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Brendan Goldsmith: Dublin Institute of Technology
Luigi Salce: Dipartimento di Matematica

A chapter in Rings, Polynomials, and Modules, 2017, pp 237-255 from Springer

Abstract: 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 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 ̃ $$\widetilde{\mathrm{ent}}$$ .

Keywords: Abelian groups; Endomorphism rings; Finite topology; Intrinsic algebraic entropy; 20K30; (endomorphisms); and; 37A35; (entropy) (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-65874-2_12

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