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  • Author: V. P. ELIZAROV x
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We describe the classes of all modules and all rings such that any system of linear equations over them is solvable if and only if it is concordant with linear relations.

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) 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.

Necessary conditions for solvability of systems of linear equations over associative rings are considered. In some cases, these conditions are also sufficient for solvability of systems.

Abstract

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.