---- > [!definition] Definition. ([[quotient module]]) > Let $R$ be a [[ring]], $M$ an $R$-[[module]], $N \subset M$ a [[submodule]]. Form $M / N$ as a [[quotient group]] and let $R$ [[module|act]] on it via $(r, m+N) \mapsto rm + N.$ > This operation is [[well-defined]] and turns $M /N$ into an $R$-[[module]], called the **quotient of $M$ by $N$**. ^definition > [!basicexample] > If $R$ is a [[ring]] and $I$ is a (two-sided) [[ideal]] of $R$, then all three of $I$, $R$, and the [[quotient ring]] $R / I$ are $R$-[[module|modules]]: $R$ as the special case $S=R, \alpha=\id$ in [[module induced by a ring homomorphism]], $I$ as a [[submodule]] of $R$, and $R / I$ as the resulting quotient. The [[ring|rings]] $R$ and $R / I$ are in fact $R$-[[algebra]] if $R$ is [[commutative ring|commutative]]. > > [!equivalence] Derivation. > ![[characterization of quotienting a module#^theorem]] ^equivalence ---- #### ---- #### 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 > ```