----- > [!proposition] Proposition. ([[finitely generated module over integral domain is torsion iff annihilator is nontrivial]]) > Let $R$ be an [[integral domain]] and $M$ a [[submodule generated by a subset|finitely generated]] $R$-[[module]]. Then $M$ is a [[torsion element of a module|torsion module]] if and only if its [[annihilator of a module|annihilator]] is nonzero: $\text{Ann}_{R}(M) \neq 0.$ ^proposition > [!proof]- Proof. ([[finitely generated module over integral domain is torsion iff annihilator is nontrivial]]) > ~ > > $\to$. Suppose $M$ is not a [[torsion element of a module|torsion module]]; let us fix some $m_{*} \in M$ that is not a [[torsion element of a module|torsion]] element. Then $r m_{*} \neq 0$ for all nonzero $r \in R$, meaning $\text{Ann}_{R}(M)$ — the set of [[ring]] elements that kill *every* $m \in M$ — must be trivial. > > $\leftarrow.$ Suppose $\text{Ann}(M)=0$. We want to witness the existence of a non-torsion element of $M$. Because $M$ is finitely generated, i.e., $R^{\oplus n} \xrightarrow{\pi} M \to 0 \text{ is exact for some (surjection) }\pi,$we have $m \in M$ iff $m$ takes on the form $m=\sum_{i=1}^{n} r_{i} \pi(\boldsymbol e_{i})$, where the $\boldsymbol e_{i}$ denote the [[free module|standard basis]] of $R^{\oplus n}$ (see [[submodule generated by a subset]]) and $r_{i}$ are coefficients from $R$. > > Using [[every spanning set reduces to maximal linearly independent set]], we may obtain a (finite, [[maximal]]) [[linearly independent]] subset $\{ \boldsymbol a_{1},\dots, \boldsymbol a_{\ell}\}$ of $M$, where each $\boldsymbol a_{j}=\pi(\boldsymbol e_{i})$ for some $i \in [n]$. Pick any set $\{ r_{i} \}_{i \in [\ell]}$ of scalars from $R$ and define $m':= \sum_{i=1}^{\ell} r_{i} \boldsymbol a_{i}.$ > Let $s \in R$ be nonzero but otherwise arbitrary. Then assume $sm'=s\sum_{i=1}^{\ell} r_{i} \boldsymbol a_{i}=\sum_{i=1}^{\ell} (r_{i} s) \boldsymbol a_{i}=0.$ > [[linearly independent|Linear independence]] says that $r_{1}s=\cdots=r_{\ell}s=0$. But $R$ is an [[integral domain]] and $s \neq 0$, thus $r_{1}=\cdots=r_{\ell}=0$. In summary, $sm'=0 \iff \{ r_{i} \}_{i \in [\ell]}=\{ 0 \} \iff m'=0.$ > It follows that $m'$ is not a [[torsion element of a module|torsion element]] of $M$. > > > > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```