---- > [!definition] Definition. ([[ideal]]) > Let $R$ be a [[ring]]. A [[subgroup]] $I$ of $(R,+)$ is a **left-ideal** of $R$ if $rI \subset I$ for all $r \in R$. It is a **right-ideal** if $Ir \subset I$ for all $r \in R$. If $I$ is both a left- and right- ideal, then we just call it an **ideal**. > The [[ideal]] $\langle 1 \rangle=R$ is called the **unit ideal**, as it is the only ideal containing [[unit|units]]. Indeed, if $I \subset R$ is an [[ideal]] and $u \in R^{*}$ is a [[unit]], then[^1] $u \in I \iff I=\langle 1 \rangle. $ > [!note] Remark. > Ideals are nearly [[subring|subrings]]: they are [[subgroup|subgroups]] that are closed under multiplication. However, in general $1_{R} \notin I$; indeed, if $1_{R} \in I$ then by definition $r 1_{R} \in I$ for all $r \in R$, implying $I=R$! > In fact, usually we care more about [[ideal|ideals]] than [[subring|subrings]]: ideals are the [[kernel of a ring homomorphism|kernels]] of [[ring homomorphism|ring homomorphisms]], and Aluffi defines the notion of [[ideal generated by a subset]] before the analog for [[subring|subrings]]. ^note ---- #### [^1]: $\impliedby$ is immediate. For $\implies$, simply observe that if $u \in I$ is a unit, then $Iu ^{-1} \subset I$ means $1 \in I$. ---- #### 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 > ```