show/hide this revision's text 2 Clarification. I'm not really sure what was proved by Gauss and what later.

A classic example for B is the theorem by (proved using Gauss, ' lemma) usually stated as: if $R$ is a unique factorization domain, then so is the ring of polynomials $R[x]$. Now, $R$ is a UFD iff it is a GCD-domain with no infinite descending chain of proper divisors , (ACCP), and

  • if $R$ is a GCD-domain, then so is $R[x]$,

  • if $R$ has no infinite chain of divisorssatisfies ACCP, then neither so does $R[x]$.
show/hide this revision's text 1

A classic example for B is the theorem by Gauss, usually stated as: if $R$ is a unique factorization domain, then so is $R[x]$. Now, $R$ is a UFD iff it is a GCD-domain with no infinite descending chain of proper divisors, and

  • if $R$ is a GCD-domain, then so is $R[x]$,

  • if $R$ has no infinite chain of divisors, then neither does $R[x]$.