Proving an ideal is prime

Mar 14, 2025 10:24 PM

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.

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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.