[[Module theory MOC]]
# Torsion

Let $M$ be a [[Module]] over a ring $R$.
A **torsion element** $t \in R$ is an element that yields zero when multiplied by some non-[[Zero-divisor]] $\lambda \in R$, i.e. $\lambda t = 0$.
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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#state/tidy | #lang/en | #SemBr