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

A commutative ring AA is said to be binomial   if AA is torsion-free (as a ZZ-module) and the element a(a−1)(a−2)⋯(a−n+1)/n!a(a−1)(a−2)⋯(a−n+1)/n! of A⊗ZQA⊗ZQ lies in AA for every a∈Aa∈A and every positive integer nn. 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 λλ-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 λλ-ring AA whose Adams operations are all the identity on AA. This allows us to construct a right adjoint BinU for the inclusion from binomial rings to rings which has several applications in commutative algebra and number theory. For example, there is a natural BinU(A)-algebra structure on the universal λλ-ring Λ(A)Λ(A), and likewise on the abelian group of multiplicative AA-arithmetic functions. Similarly, there is a natural BinU(A)-module structure on the abelian group 1+a1+a for any ideal aa in AA with respect to which AA is complete.