Endomorphism rings with different rank-entropy supports. (Q2890806)

This paper unites three significant areas of algebra. One is the algebraic rank entropy of torsion-free Abelian groups, that is, the chaotic behaviour of endomorphisms acting on finitely generated subgroups; the second is the structure of rings admitting different topologies; and the third is the realization of topological rings as the endomorphism rings of Abelian groups.NEWLINENEWLINE The main result is the construction of non-isomorphic \(\aleph_1\)-free groups whose endomorphism rings are algebraically, but not topologically isomorphic, one having rank entropy zero and the other rank entropy infinity. The authors also show that if two such groups have algebraically isomorphic endomorphism rings, then with respect to the finite topology, they must have the same rank entropy.