Implications of a system of linear equations over a module

Abstract

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 η ↓ = γ ↓ over a module from L ( R ) the system of equations A ξ ↓ = 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.