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 Fη↓ = γ↓ over a module from L(R) the system of
equations Aξ↓ = β↓ is its D-implication if and only if T(F, γ↓) = (AD,β↓) 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.
A system of linear equations over a ring R is called factorially solvable if for any proper ideal I of R its factorsystem is solvable over the ring R/I. A ring is called factorially solvable if any factorially solvable system over this ring is solvable. In this article it is shown that any decomposable ring is factorially solvable, a commutative principal ideal domain is factorially solvable if and only if it is subdirectly indecomposable, and that a finite commutative ring is factorially solvable if and only if it is not local.