Example of injective ring homomorphism that is not flat.

Question

What is a good example of an injective ring homomorphism $f: A \rightarrow B$ that is not flat, i.e. such that $B$ is not a flat $A$-module? Examples of non-flat modules I've seen usually involve an element $a \in A$ such that $aB = 0$, hence not injective, like $\mathbb{Z}_2$ as a $\mathbb{Z}$-module.

Answer 1

You can use the examples you know to construct an injective example.

For instance, take the morphism $f:\mathbb{Z}\to \mathbb{Z}\times \mathbb{Z}_2:n\to (n, \overline{n})$. Then $f$ is injective, but $\mathbb{Z}\times \mathbb{Z}_2$ is not a flat $\mathbb{Z}$-module.