KRULL DIMENSION AND UNIQUE FACTORIZATION IN HURWITZ POLYNOMIAL RINGS

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 �� from N0 to N. The trivial case when ��(i) = 1 for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ��(i) = i! for all i. For this case, it gives the well-known Hurwitz polynomial ring RH[x]. In this paper, we study Krull dimension and unique factorization in RH[x]. We show in general that dimR �� dimRH[x] �� 2 dimR + 1. When the ring R is Noetherian we prove that dimR �� dimRH[x] �� dimR + 1. A condition for the ring R is also given in order to determine whether dimRH[x] = dimR or dimRH[x] = dimR+1 in this case. We show that RH[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. ? 2017 Rocky Mountain Mathematics Consortium.