Going up theorem

Statement

This result is sometimes called going up and sometimes lying over and going up. It is a stronger version of lying over.

Suppose $f:R\to S$ is an injective homomorphism of commutative unital rings, such that $S$ is an integral extension of $R$. Suppose $P$ is a prime ideal of $R$, and $Q_1$ is an ideal of $S$ such that $f^{-1}(Q_1)\subset P$. Then, there exists a prime ideal $Q$ containing $Q_1$, such that $f^{-1}(Q)=P$.

Proof

This follows from lying over, applied to the injective map $R/f^{-1}(Q_1)\to S/Q_1$.