Let $R$ be a ring and $A, B, C$ $R$-algebras such that there is an injective $R$-algebra Homomorphism $i: B \hookrightarrow C$. Is it true that the induced map $j: A \otimes_R B \to A \otimes_R C$ is injective?
As @Mohan has pointed out the answer to this is negative. Does the situation change if we replace $R$ by a field $k$?