Chain conditions on prime ideals of commutative rings which holds in the polynomial ring also

Question

If $R$ is a commutative ring with unity , and every prime ideal of $R$ is finitely generated , then every prime ideal of $R[x]$ is also finitely generated as $R$ is Noetherian . However , even if the prime ideals of $R$ satisfies ACC (ascending chain condition ) , $R$ may not be Noetherian . But my question is this : If the prime ideals of $R$ satisfy ACC , then do the prime ideals of $R[x]$ also satisfy ACC ? What if the prime ideals of $R$ satisfy DCC , is the same chain condition lifted to $R[x]$ as well ?