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Binomial rings, integer-valued polynomials, and \(\lambda\)-rings. (English) Zbl 1100.13026

If \(a\in A\), a commutative ring, and \(n\) is a positive integer, then the binomial coefficient \(\binom an\) is an element of \(A\otimes_\mathbb Z \mathbb Q\) but not necessarily of \(A\). \(A\) is called a binomial ring if \(\binom an\in A\) for all \(a\in A\) and all positive integers \(n\). Binomial rings have applications to nilpotent groups, integer-valued polynomials in \(\mathbb Q[X]\), Witt vectors and \(\lambda\)-rings. In this well-written paper, the author characterizes binomial rings and their homomorphic images in several ways, and presents applications in commutative algebra and number theory.
Binomial rings were first defined in 1969 by P. Hall [Nilpotent groups. Notes of lectures given at the Canadian Mathematical Congress, summer seminar, University of Alberta, Edmonton, 12–30 August, 1957. Queen Mary College Mathematics Notes. London: Queen Mary College (1969; Zbl 0211.34201)] in connection with his groundbreaking work in the theory of nilpotent groups.

MSC:

13F25 Formal power series rings
13F35 Witt vectors and related rings

Citations:

Zbl 0211.34201
Full Text: DOI

References:

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