# R[x] UFD, then R UFD

1. Nov 23, 2009

### math_grl

1. The problem statement, all variables and given/known data

Assuming R is an integral domain. If the polynomial ring of one variable, R[x], is a unique factorization domain, then R is a unique factorization domain.

3. The attempt at a solution

Should be straightforward...so much so that I don't know how to start...probably with a homomorphism...

2. Nov 23, 2009

### rasmhop

An integral domain is a UFD iff every reducible element has a unique factorization. So consider an arbitrary reducible element $a \in R$. Then a is an element of R[x] of degree 0, and since R[x] is a UFD a must have a unique factorization in R[x]. Can you from this unique factorization conclude that it has a unique factorization in R? (HINT: If $a = a_1a_2\cdots a_n$ is $a_i \in R$ for all i?)

3. Nov 23, 2009