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 $$\text{Int\,}R$$.

MSC:

 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
Full Text:

References:

 [1] Cahen, P.-J.; Chabert, J.-L., Integer-valued polynomials, Math. surveys monogr., (1997), Amer. Math. Soc. Providence, RI [2] Flath, D.E., Introduction to number theory, (1989), Wiley New York · Zbl 0651.10001 [3] Frisch, S., Polynomial separation of points in algebras, (), 253-259 · Zbl 1092.13027 [4] Goodearl, K.R.; Warfield, R.B., An introduction to noncommutative Noetherian rings, (2004), Cambridge University Press New York · Zbl 1101.16001 [5] Hurwitz, A., Vorlesungen über die zahlentheorie der quaternionen, (1919), J. Springer Berlin · JFM 47.0106.01 [6] Lam, T.Y., A first course in noncommutative rings, (2001), Springer-Verlag New York · Zbl 0980.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.