Module induced from projective is projective

Question

Let $A,B$ be rings such that $A$ is $B$-module, $P$ be projective $B$-module. I want to prove that $A\otimes_B P$ is projective. I have that $\mathrm{Hom}_A(A\otimes_B P,M) \simeq \mathrm{Hom}_B(P,\mathrm{Hom}_B(A,M))$ so if $A$ is projective than $\mathrm{Hom}_A(A\otimes_B P,M)$ is exact and $A\otimes_B P$ is projective. But it's not always true and i can't get what to do..

Answer 1

It's much simpler than what you tried: $P$, as a $B$-module, is a direct summand of some free $B$-module $F$, hence $A\otimes_B P$, as an $A$-module, is a direct summand of the free $A$-module $A\otimes_B F$.