Binomial rings, integer-valued polynomials, and [formula omitted]-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.