For a unitary algebra $A$, i.e. an algebra with identity(if necessary we can assume it is commutative), let $M$ denote an $A$-module of finite type. It is reflexive, if $$M^{**}\cong M,$$ where $M^*=\mathrm{Hom}_A(M,A)$.
Q: For a finite type $A$-module $M$ which is reflexive can we construct a short exact sequence $$E_0\to E_1\to M^*\to0,$$ where $E_0$ is a free $A$-modules and $E_1$ is a torsion-free module ?