Prime Ideals

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A prime ideal of a ring \(R \) is proper ideal \(P \) such that whenever \(xy\in P \) for \(x,y \in R \), \(x\in P \) or \(y \in P \).
An ideal \(P \) is prime if and only if \( R\backslash P\) is closed under multiplication.
Maximal implies prime but not conversely.

An ideal \(P\) is prime if and only if \(R \P \) is closed under multiplication.

If \(R\) is a domain, \(S\) is a ring and \(f : R \to S\) is a ring homomorphism, then \(\text{Ker}(f)\) is a prime ideal.

The ideal \(I\) is prime if and only if \(R/I\) is an integral domain.