---- > [!definition] Definition. ([[principal ideal]]) > Let $R$ be a [[ring]] and $a \in R$. Then the subset $Ra \subset R$ is a left-[[ideal]] of $R$ ($rRa=\overbrace{(rR)}^{\subset R}a \subset Ra$) and the subset $aR \subset R$ is a right-[[ideal]] of $R$. When $Ra=aR$ (e.g., when $R$ is [[commutative ring|commutative]]) we call it the **principal ideal generated by $a$** and write $\langle a \rangle:=Ra=aR$. > ^definition > [!basicexample] > - The zero-ideal $\{ 0 \}=(0)$ and the whole [[ring]] $R=(1)$ are both principal ideals. ^basic-example ---- #### ---- #### 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 > ```