Does restriction of scalars preserve projectivity?

Question

Suppose $A \to B$ is a ring homomorphism. If $P$ is a projective $B$-module, is $P$ projective as an $A$-module?

I can prove this with the added hypothesis that $B$ is projective as an $A$-module, but is it true more generally?

Answer 1

It's not true. Take $\mathbf{Q}$ as a $\mathbf{Q}$-module and a $\mathbf{Z}$-module.