[Math] example of a flat but not faithfully flat ring extension

Question

I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. Can someone help me with it? thanks

Answer 1

Take $f: A \rightarrow B$ to be a ring homomorphism such that the corresponding morphism of affine schemes $\operatorname{Spec}B \rightarrow \operatorname{Spec}A$ is not surjective, but only flat. There is an easy way to do this: Remember that localizing a ring $R$ in a multiplicative subset $S$ gives a flat ring homomorphism $R \rightarrow S^{-1}R$. However, this ring homomorphism is faithfully flat iff $S^{-1}R=R$.