Example of a universally injective ring morphism which is not faithfully flat

Question

Is there some classic example of an universally injective ring morphism which is not a faithfully flat morphism?

I was not able to find it in any commutative algebra book and neither around here.

Answer 1

Here is a simple example. Take a ring $R$ and a non-flat module $M$. Then, take $S=R\oplus M$ with the ring structure given by $M^2=0$. So, $(a,m)\cdot (b,n)=(ab, an+bm)$. Then the map $R\to S $ is split and thus universally injective. But $S$ is not flat over $R$.