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- W2743134871 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 $dim R leq dim R_H[x] leq 2dim R +1$. 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." @default.
- W2743134871 created "2017-08-17" @default.
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- W2743134871 date "2017-08-01" @default.
- W2743134871 modified "2023-09-23" @default.
- W2743134871 title "Krull dimension and unique factorization in Hurwitz polynomial rings" @default.
- W2743134871 doi "https://doi.org/10.1216/rmj-2017-47-4-1317" @default.
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