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2017 Krull dimension and unique factorization in Hurwitz polynomial rings
Phan Thanh Toan, Byung Gyun Kang
Rocky Mountain J. Math. 47(4): 1317-1332 (2017). DOI: 10.1216/RMJ-2017-47-4-1317

Abstract

Let $R$ be a commutative ring with identity, and let $R[x]$ be the collection of polynomials with coefficients in~$R$. We observe that there are many multiplications in $R[x]$ such that, together with the usual addition, $R[x]$ becomes a ring that contains $R$ as a subring. These multiplications belong to a class of functions $\lambda $ from $\mathbb {N}_0$ to $\mathbb {N}$. The trivial case when $\lambda (i) = 1$ for all $i$ gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when $\lambda (i) = i!$ for all $i$. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we study Krull dimension and unique factorization in $R_H[x]$. We show in general that $\dim R \leq \dim R_H[x] \leq 2\dim R +1$. When the ring $R$ is Noetherian we prove that $\dim R \leq \dim R_H[x] \leq \dim R+1$. A condition for the ring $R$ is also given in order to determine whether $\dim R_H[x] = \dim R$ or $\dim R_H[x] = \dim R +1$ in this case. We show that $R_H[x]$ is a unique factorization domain, respectively, a Krull domain, if and only if $R$ is a unique factorization domain, respectively, a Krull domain, containing all of the rational numbers.

Citation

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Phan Thanh Toan. Byung Gyun Kang. "Krull dimension and unique factorization in Hurwitz polynomial rings." Rocky Mountain J. Math. 47 (4) 1317 - 1332, 2017. https://doi.org/10.1216/RMJ-2017-47-4-1317

Information

Published: 2017
First available in Project Euclid: 6 August 2017

zbMATH: 06790016
MathSciNet: MR3689956
Digital Object Identifier: 10.1216/RMJ-2017-47-4-1317

Subjects:
Primary: 13B25, 13C15, 13E05, 13F15, 13N99

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 4 • 2017
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