Noetherian module and noetherian ring If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?

A noetherian $R$-module has all its submodules finitely generated – in particular it has to be finitely generated as an $R$-module. If $R[x]$ were finitely generated, it would be generated by a finite number of polynomials as an $R$-module, so that polynomials would have a bounded degree.
