Korean J. Math. Vol. 30 No. 3 (2022) pp.425-431
DOI: https://doi.org/10.11568/kjm.2022.30.3.425

Factorization in the ring $h(\mathbb{Z}, \mathbb{Q})$ of composite Hurwitz polynomials

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Dong Yeol Oh
Ill Mok Oh


Let $\mathbb{Z}$ and $\mathbb{Q}$ be the ring of integers and the field of rational numbers, respectively. Let $h(\mathbb{Z}, \mathbb{Q})$ be the ring of composite Hurwitz polynomials. In this paper, we study the factorization of composite Hurwitz polynomials in $h(\mathbb{Z}, \mathbb{Q})$. We show that every nonzero nonunit element of $h(\mathbb{Z}, \mathbb{Q})$ is a finite $*$-product of quasi-primary elements and irreducible elements of $h(\mathbb{Z}, \mathbb{Q})$. By using a relation between usual polynomials in $\mathbb{Q}[x]$ and composite Hurwitz polynomials in $h(\mathbb{Z}, \mathbb{Q})$, we also give a necessary and sufficient condition for composite Hurwitz polynomials of degree $\leq 3$ in $h(\mathbb{Z}, \mathbb{Q})$ to be irreducible.

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[1] A. Benhissi, Ideal structure of Hurwitz series ring, Contrib. Alg. Geom. 48 (1997) 251–256. Google Scholar

[2] A. Benhissi and F. Koja, Basic properties of Hurwitz series rings, Ric. Mat. 61 (2012) 255–273. Google Scholar

[3] L. Fuchs, On quasi-primary ideals, Acta. Sci. Math. (Szeged), 11 (1947) 174–183. Google Scholar

[4] W.F. Keigher, Adjunctions and comonads in differential algebra, Pacific J. Math. 59 (1975) 99–112. Google Scholar

[5] W.F. Keigher, On the ring of Hurwitz series, Comm. Algebra 25 (1997) 1845–1859. Google Scholar

[6] J.W. Lim and D.Y. Oh, Composite Hurwitz rings satisfying the ascending chain condition on principal ideals, Kyungpook Math. J. 56 (2016) 1115–1123. Google Scholar

[7] J.W. Lim and D.Y. Oh, Chain conditions on composite Hurwitz series rings, Open Math. 15 (2017) 1161–1170. Google Scholar

[8] Z. Liu, Hermite and PS-rings of Hurwitz series, Comm. Algebra 28 (2000) 299–305. Google Scholar

[9] D.Y. Oh and Y.L. Seo, Irreducibility of Hurwitz polynomials over the ring of integers, Korean J. Math. 27 (2019) 465–474. Google Scholar