## 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
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### References:

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