Here is the question:
Let F be a field, and R a commutative ring with identity. If φ:F[x]→R is a surjective ring homomorphism such that ker φ = (m(x)) for some m(x) ∈ F [x], prove that φ induces an isomorphism F[x]/(m(x)) →∼ R.
So based on my understanding, to prove isomorphism I must show the transformation is injective and surjective. We already know that it is surjective so I just need to show that it is also one-to-one.
Since the kernal is m(x), we know the kernal is non-trivial. So I'm guessing for that reason φ cannot be injective, but somehow F[x]/(m(x)) →∼ R is injective but I am stuck with that.
I am thinking that I need to show that the kernal of F[x]/m(x)) is 0. But the question says it is m(x). But m(x) cannot be 0 since it is the denominator.