For an extension of commutative rings $R \subset S$ , the extension is said to satisfy
Going Up (GU) property if for every chain of prime ideals $P\subseteq P_0$ of $R$ with $P= Q \cap R$ for some prime ideal $Q$ of $S$ , there is a prime ideal $Q_0$ of$S$ such that $ Q \subseteq Q_0$ and $P_0= Q_0 \cap R$.
Now my question is obviously we know this condition is true if $R$ is an integral extensions of $S$. But I am searching for a non-integral extension which will satisfy the going up property. Any help will be appreciated. Thanks in advance