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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.
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참고문헌 (19)
참고문헌 신청
A. Benhissi / 2007 / Ideal structure of Hurwitz series rings / Beitrage Algebra Geom. 48(1):251~256
A. Benhissi / 2011 / Factorization in Hurwitz series domain / Rend. Circ. Mat. Palermo (2) 60(1-2):69~74
A. Benhissi / 2011 / PF and PP-properties in Hurwitz series ring / Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102)(3):203~211
A. Benhissi / 2014 / Chain condition on annihilators and strongly Hopfian property in Hurwitz series ring / Algebra Colloq. 21(4):635~646
A. Benhissi / 2012 / Basic properties of Hurwitz series rings / Ric. Mat. 61(2):255~273
A. Benhissi / 2011 / Factorization in Hurwitz series domain / Rend. Circ. Mat. Palermo (2) 60(1-2):69~74
A. Benhissi / 2011 / PF and PP-properties in Hurwitz series ring / Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102)(3):203~211
A. Benhissi / 2014 / Chain condition on annihilators and strongly Hopfian property in Hurwitz series ring / Algebra Colloq. 21(4):635~646
A. Benhissi / 2012 / Basic properties of Hurwitz series rings / Ric. Mat. 61(2):255~273
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