Proving an ideal is prime

Suppose $ϕ : R \to S$ is a ring homomorphism, and $S$ has no (nonzero) zero-divisors. Prove from the definitions that $\ker (ϕ)$ is a prime ideal.

View

L A T E X

code

Suppose $\phi:R\to S$ is a ring homomorphism, and $S$ has no (nonzero) zero-divisors. Prove from the definitions that $\ker(\phi)$ is a prime ideal.