Let $A$ be a commutative ring with unity and $M$ an invertible $A$-module. Suppose that we have a surjection $\varphi: A^{2} \to M$ with $\varphi(e_1)=m_1, \varphi(e_2)=m_2$ and suppose further that we have $a_1,a_2 \in A$ such that $$a_1m_2 - a_2m_1=0.$$
Does this imply that the module $M\cong A$?