Module theory MOC

Torsion

Let 𝑀 be a Module over a ring 𝑅. A torsion element 𝑡 ∈𝑅 is an element that yields zero when multiplied by some non-Zero-divisor 𝜆 ∈𝑅, i.e. 𝜆𝑡 =0. This is a strong deviation from the behaviour of a vector space, as torsion elements cannot exist for a module over a field, where scalar multiplication is injective, hence vector spaces are torsion-free. A torsion module consists of only torsion elements. Given a module, the set of all torsion elements forms the Torsion submodule.

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