Integer-valued polynomials over quaternion rings. (English) Zbl 1219.16027

Let \(R=\mathbb ZQ\) be the ring of quaternions with integer coefficients. Then its division ring of fractions is equal to \(D=\mathbb Q\otimes_{\mathbb Z}R\). Denote by \(\text{Int\,}R\) the ring of all polynomials \(f\) with coefficients in \(D\) such that \(f(R)\subseteq R\). For each positive integer \(n\) denote by \(I_n\) the ideal of polynomials in \(R[X]\) such that \(f(a)\equiv 0\bmod n\) for all \(a\in R\).
It is shown that \(\text{Int\,}R \) is not a Noetherian ring. There are found explicit forms of finitely many generators from \(\mathbb Z[X]\) of each ideal \(I_n\).
There are given some consequences for number theory. For example, if \(u\) is a positive integer, \(u\equiv 1,2,3,5\text{ or }6\bmod 8\), then there exist coprime integers \(y,z,w\) such that \(u=y^2+z^2+w^2\). There is also given a classification of maximal ideals of \(\text{Int\,}R\).


16S36 Ordinary and skew polynomial rings and semigroup rings
16K20 Finite-dimensional division rings
11E25 Sums of squares and representations by other particular quadratic forms
11R52 Quaternion and other division algebras: arithmetic, zeta functions
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16D25 Ideals in associative algebras
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