Implications of a system of linear equations over a module

We describe the class $L(R)$ of all left modules over a ring $R$ such that for any matrix $D$ over $R$ and any solvable system of equations
$$ F\eta^\downarrow=\gamma^\downarrow $$
over a module from $L(R)$ the system of equations
$$ A\xi^\downarrow=\beta^\downarrow $$
is its $D$-implication if and only if
$$ T(F,\gamma^\downarrow)=(AD,\beta^\downarrow) $$
for some matrix $T$. If $R$ is a quasi-Frobenius ring, then $L(R)$ contains the subclass of all faithful $R$-modules. A criterion for a system of equations over a module from $L(R)$ to be definite is obtained.