$R$ is a commutative ring, show $xR[x]$ is prime iff $R[x]$ is an integral domain

Question

$R$ is a commutative ring, show $xR[x]$ is prime iff $R[x]$ is an integral domain and show $xR[x]$ is maximal iff $R[x]$ is a field.

Now, I'm trying to use the theorem that a quotient ring is an integral domain iff the ideal is prime, but this situation is a little different. Once I get one of the arguments down then i'm sure the only will follow by adopting a common strategy of how to turn the proof of why the quotient ring that is formed by using a maximal ideal is a field and adapting it to this situation. Any help is appreciated!!

Answer 1

Consider $f:R[X]\rightarrow R$ defined by $f(P)=P(0)$, it is a morphism and its kernel is $XR[X]$.