Rocky Mountain Journal of Mathematics

Krull dimension and unique factorization in Hurwitz polynomial rings

Phan Thanh Toan and Byung Gyun Kang

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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.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 4 (2017), 1317-1332.

Dates
First available in Project Euclid: 6 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1501984951

Digital Object Identifier
doi:10.1216/RMJ-2017-47-4-1317

Mathematical Reviews number (MathSciNet)
MR3689956

Zentralblatt MATH identifier
06790016

Subjects
Primary: 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13C15: Dimension theory, depth, related rings (catenary, etc.) 13E05: Noetherian rings and modules 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05] 13N99: None of the above, but in this section

Keywords
Hurwitz polynomial Krull dimension Noetherian ring polynomial ring unique factorization

Citation

Toan, Phan Thanh; Kang, Byung Gyun. Krull dimension and unique factorization in Hurwitz polynomial rings. Rocky Mountain J. Math. 47 (2017), no. 4, 1317--1332. doi:10.1216/RMJ-2017-47-4-1317. https://projecteuclid.org/euclid.rmjm/1501984951


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