Abstract:

In this paper, we study the existence of non-torsion solutions of a homogeneous linear system over a commutative ring. More precisely, we determine the minimal positive integer

*n* such that any homogeneous systems of

*m* equations with

*n* variables over a given ring

*R* gives a non-torsion solution, i.e. a solution

*x* =(

*x*_{1},

*x*_{2}, . . . ,

*x*_{n})such that at least one coordinate

*x*_{i} is not a zero-divisor. We proved that over Noetherian rings, a non-trivial lower bound to the minimal number can be guaranteed via the use of primary decomposition. We also consider the number of generators of ideals in

*R* and the localisations of

*R*. For some classes of Noetherian rings, such as principal ideal rings and reduced rings, we show that such minimal number exists.

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