Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2018; 55(2): 625-631

Online first article January 12, 2018      Printed March 30, 2018

https://doi.org/10.4134/BKMS.b170201

Copyright © The Korean Mathematical Society.

Krull dimension of Hurwitz polynomial rings over Pr\"ufer domains

Le Thi Ngoc Giau, Phan Thanh Toan

Ton Duc Thang University, Ton Duc Thang University

Abstract

Let $R$ be a commutative ring with identity and let $R[x]$ be the collection of polynomials with coefficients in $R$. There are a lot of multiplications in $R[x]$ such that together with the usual addition, $R[x]$ becomes a ring that contains $R$ as a subring. These multiplications are from 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 completely determine the Krull dimension of $R_H[x]$ when $R$ is a Pr\"ufer domain. Let $R$ be a Pr\"ufer domain. We show that $\dim R_H[x] = \dim R+1$ if $R$ has characteristic zero and $\dim R_H[x] = \dim R$ otherwise.

Keywords: Hurwitz polynomial, Krull dimension, polynomial ring, Pr\"ufer domain

MSC numbers: 13B25, 13C15, 13F05