Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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KRULL DIMENSION OF HURWITZ POLYNOMIAL RINGS OVER PRÜFER DOMAINS

  • Le, Thi Ngoc Giau (Faculty of Mathematics and Statistics Ton Duc Thang University) ;
  • Phan, Thanh Toan (Faculty of Mathematics and Statistics Ton Duc Thang University)
  • Received : 2017.03.07
  • Accepted : 2017.07.21
  • Published : 2018.03.31
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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. 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 $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ 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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References

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