---- > [!definition] Definition. ([[simple module]]) > Let $R$ be a [[ring]]. A nonzero $R$-[[module]] $M$ is **simple** (or **irreducible**) if its only [[submodule|submodules]] are $\{ 0 \}$ and $M$. ^definition > [!basicproperties] > - Simple modules are [[cyclic module|cyclic]] > ^properties > [!proof] Proof of basic properties. > (If $R=\{ 0 \}$ then there is nothing to talk about, so assume $R$ is nonzero.) > Let $M$ be a simple module and let $m \in M$ with $m \neq 0$. The statement that $M$ is [[cyclic module|cyclic]] is equivalent to saying that there exists a [[surjection|surjective]] [[linear map]] from $R^{1}$ to $M$. Consider the [[linear map]] $\varphi:R^{1} \to M$ obtained by sending $1 \mapsto m$ and homomorphically extending: explicitly, $\varphi$ must be defined $\varphi(s)=s \varphi(1)=sm.$ The image of $\varphi$ must be a [[submodule]] of $M$; since $\varphi$ is nonzero, it must in fact be $M$ itself. So $\varphi$ is the required [[surjection]]. ^proof ---- #### ---- #### 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 > ```