Representation of a tetrad

A complete description is given herein of finitely generated torsionless modules over the ring
$$ A=\{(a_1,a_2,a_3,a_4)\mid a_i\in A_i,i=1,\dots,4,\ a_1\varepsilon_1=a_2\varepsilon_2=a_3\varepsilon_3=a_4\varepsilon_4\}, $$
where $A_1$, $A_2$, $A_3$, $A_4$ are local Dedekind rings with the same residue field $k$, and $\varepsilon_i$ is the homomorphism of $A_i$ onto $k$.