Example 8

Let $B$ be a ring and $\rho : A \to B$ a homomorphism. If $E$ is a flat $A$-module, the $B$-module $E_{(B)} = B \otimes_A E$ is flat. Indeed, let $u : N' \to N$ be an injective homomorphism of right $B$-modules; then $u \otimes_B 1_{E_{(B)}}$ is canonically identified with the homomorphism $u \otimes_A 1_E : N' \otimes_A E \to N \otimes_A E$, which is injective if $E$ is flat.