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# Contents

## Definition

### For modules over rings

Given a ring $R$, an element $m$ in an $R$-module $M$ is torsion element if there is a nonzero element $r$ in $R$ such that $r m=0$. In constructive mathematics, given a ring $R$ with a tight apartness relation $\#$, an element $m$ in an $R$-module $M$ is a torsion element if there is a element $r$ in $R$ such that $r \# 0$ and $r m=0$.

A torsion module is a module whose elements are all torsion. A torsion-free module is a module whose elements are not torsion, other than $0$.

More generally, given an ideal $\mathfrak{a} \subset R$ then an $\mathfrak{a}$-torsion module is one all whose elements are annihilated by some power of elements in $\mathfrak{a}$.

### For $\infty$-modules over $E_\infty$-rings

Let $A$ be an E-∞ ring and $\mathfrak{a} \subset \pi_0 A$ a finitely generated ideal of its underlying commutative ring.

###### Definition

An $A$-∞-module $N$ is an $\mathfrak{a}$-torsion module if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.

###### Proposition
$A Mod_{\mathfrak{a}tor} \hookrightarrow A Mod$

is co-reflective and the co-reflector $ʃ_{\mathfrak{a}}$ – the torsion approximation – is smashing.

###### Proposition

For $N \in A Mod_{\leq 0}$ then torsion approximation, prop. , intuced a monomorphism on $\pi_0$

$\pi_0 ʃ_{\mathfrak{a}} N \hookrightarrow \pi_0 N$

including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.

## References

Last revised on May 19, 2022 at 06:09:45. See the history of this page for a list of all contributions to it.