Thread: Prime Ideal Integral domain

Let R be a commutative ring with unity. Show that the ideal generated by <x>={xf(x)|f(x) is a polynomial in R[x]} is a prime ideal in R[x] if and only if R is an integral domain.

For the forward direction I tried to use fact that an ideal N of R is prime if and only if R/N is an integral domain. But got stuck after that.

And I am completely clueless about the backwards direction. Sorry. Any help would be appreciated.

Originally Posted by curiousmuch

Let R be a commutative ring with unity. Show that the ideal generated by <x>={xf(x)|f(x) is a polynomial in R[x]} is a prime ideal in R[x] if and only if R is an integral domain.

For the forward direction I tried to use fact that an ideal N of R is prime if and only if R/N is an integral domain. But got stuck after that.

And I am completely clueless about the backwards direction. Sorry. Any help would be appreciated.

define the map by it's easy to see that is a surjective ring homomorphism and thus and the result follows.

Oh wow this way both directions are taken care of at once.