0. Prime ideals

Definition. An ideal $P$ of $R$ is prime if $P < R$ and $xy \in P$ implies $x \in P$ or $y \in P$.

Proposition. An ideal $P$ of $R$ is prime $\Leftrightarrow$ R/P is an integral domain.

Proof. What does the definition of prime say when expressed mod $P$?

Corollary. The prime ideals of $R$ are the kernels of homomorphisms of $R$ onto an integral domain.