---- Let $R$ be a [[ring]]. > [!definition] Definition. ([[annihilator of a module]]) > The **annihilator** of an $R$-[[module]] $M$ is the [[ideal]] $\text{Ann}_{R}(M):= \{ r \in R: \forall m \in M, rm=0 \}$ > of $R$. > > The **annihilator of $m \in M$** is the [[ideal]] $\text{Ann}_{R}(m)=\{ r \in R: rm=0 \}.$ ^definition > [!justification] > We should check this is indeed an [[ideal]] of $R$. Indeed, given any $s \in R$ and $r \in \text{Ann}(M)$, we have > $(sr)(m)=s(rm)=s(0)=0 \text{ for all }m \in M$ and if $r_{1},r_{2} \in \text{Ann}(M)$. The [[subgroup]] requirement is similarly easy to check. ^justification ---- #### ---- #### 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 > ```