Let $A$ be a ring and $E$ a left $A$-module. Write $\theta$ for the canonical homomorphism from $E^*\otimes_AE$ into $\text{End}_A(E)$ such that $$\theta(x^*\otimes y)(x)=\langle x,x^*\rangle y$$ for all $x^*\in E^*$ and $(x,y)\in E\times E$.
I want to show that $E$ is finitely generated projective if and only if $1_E\in\text{Im}(\theta)$.
It is clear that the condition is necessary. I am not sure how to show that it's sufficient. If the condition holds, then $E$ is clearly projective. But why would it have to be finitely generated?
Hint: If $1_E=\sum_ix_i^*\otimes y_i$, then $y_i$ generates $E$.