Torsion-free Modules

A torsion-free module is a mathematical structure in algebra where multiplying any non-zero element by a non-zero scalar never results in zero. In other words, it lacks elements that become zero solely through multiplication by non-zero scalars—a property called torsion. This concept extends from modules over rings, generalizing the idea of torsion-free abelian groups, ensuring that the structure behaves predictably under scalar operations, maintaining the integrity of its elements without "collapsing" to zero unless originally zero.