**Seminars and Colloquiums**

for the week of January 18, 2016

for the week of January 18, 2016

*SPEAKER:*

Phan Thanh Toan (POSTECH, S. Korea), Monday

*TEA TIME - (cancelled this week)
3:00 – 3:30 pm, Ayres 401
Monday, Tuesday, & Wednesday
Hosted by *

**Monday, January 18th **

ALGEBRA SEMINAR

TITLE: Hurwitz polynomial rings

SPEAKER: Phan Thanh Toan (POSTECH, S. Korea)

TIME: 2:00pm - 3:00pm

ROOM: Ayres 114

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 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 $i$. In this case, we get the well-known Hurwitz polynomial ring R_H[x].

We show in general that $\dim R \leq \dim R_H[x] \leq 2\dim R +1$.

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 (resp., a Krull domain)

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