Aperiodic rings, necklace rings, and Witt vectors

In [N. Metropolis and Gian Carlo Rota, Witt Vectors and the Algebra of Necklaces, Adv. in Math.50 (1983), 95–125], Metropolis and Rota show that the cyclotomic identity provides a linkage between the necklace ring and the ring of Witt vectors. In this paper we study this linkage. Let A be the class of rings A with the property that the additive group of A is torsion-free. We will prove that the ring of Witt vectors over A for any A ϵ A is isomorphic to a suitably defined subring of the necklace ring of the rationalization A ⊗ Q of A, alternatively to a subring of the aperiodic ring of A. A major part of this paper is concerned with a development of the properties and relationships involving the aperiodic ring, which parallels the treatment of the necklace ring given in [N. Metropolis and Gian-Carlo Rota, Witt Vectors and the Algebra of Necklaces, Adv. in Math.50 (1983), 95–125]. In addition we translate the combinatorial proof of the cyclotomic identity given by Metropolis and Rota [N. Metropolis and Gian-Carlo Rota, The cyclotomic identity, Contemp. Math.34 (1984), 19–27] into the language of species in the sense of Joyal, thereby making clear the “naturality” of the identity.