Binomial rings, integer-valued polynomials, and λ-rings

Abstract

A commutative ring A is said to be binomial if A is torsion-free (as a Z-module) and the element a(a − 1)(a − 2) · · · (a − n + 1)/n! of A ⊗ Z Q lies in A for every a ∈ A and every positive integer n. Binomial rings were first defined circa 1969 by Philip Hall in connection with his groundbreaking work in the theory of nilpotent groups. They have since had further applications to integer-valued polynomials, Witt vectors, and λ-rings. For any set X , the ring of integer-valued polynomials in Q[X ] is the free binomial ring on the set X . Thus the binomial property provides a universal property for rings of integer-valued polynomials. We give several characterizations of binomial rings and their homomorphic images.For example, we prove that a binomial ring is equivalently a λ-ring A whose Adams operations are all the identity on A. This allows us to construct a right adjoint Bin U for the inclusion from binomial rings to rings which has several applications in commutative algebra and number theory. For example, there is a natural Bin U (A)-algebra structure on the universal λ-ring Λ(A), and likewise on the abelian group of multiplicative A-arithmetic functions. Similarly, there is a natural Bin U (A)-module structure on the abelian group 1 + a for any ideal a in A with respect to which A is complete.