Let $R$ be a commutative ring with unity and let $A, B$ be $R$ algebras. In particular, $B$ is a field. Consider the tensor product $A \otimes_R B$. I have proved that the pure tensor $1 \otimes x = 0$. I think it must be the case that then $x=0 \in B$.. but I was not sure how to prove it. Any explanation is appreciated. Thank you.
Edit. $R \to A$ is an injection
Edit 2. The above is false as pointed out in the comments. The exact situation I have is the following: Let $R \to A$ an injection and $B = R_{P}$, where $R_P$ is $R$ localised at $P$ and $P$ is a minimal ideal of $R$, which makes $R_{P}$ a field.