---- Let $R$ be a [[ring]]. > [!definition] Definition. ([[unit]]) > $u \in R$ is a **left-unit** if there exists $v \in R$ such that $uv=1$; it is a **right-unit** if there exists $v \in R$ such that $vu=1$. If both, we just call $u$ **unit**. > The inverse of a (two-sided) unit is unique (see properties below); it is denoted $u ^{-1}$. ^definition > [!definition] Definition. ([[group of units of a ring]]) > The set of [[unit|units]] of a [[ring]] $R$ form a [[group]] under multiplication, called the **group of units** of **unit group** of $R$. ^definition > [!note] Remark. > A bit like the notion of [[zero-divisor]], but now we care about multiplicative identity $1$ rather than additive identity $0$. ^note > [!basicproperties] > - $u$ is a left- (resp., right-) unit if and only left- (resp., right) multiplication by $u$ is a [[surjection|surjective function]] $R \to R$ > > **Left.** Let $u$ be a left-unit; fix $v \in R$ such that $uv=1$. Then for any $b \in R$, we have $u(vb)=(uv)b=1b=b$, so left-multiplication is [[surjection|surjective]]. Conversely suppose $u$ is not a left-unit. Then by definition there is not $v \in R$ for which $uv=1$, so $1$ is not in the image of the left-multiplication function! > > **Right.** Also easy to check. > > - If $u$ is a left- (resp., right-) unit, then right (resp., left) multiplication by $u$ is [[injection|injective]]; [[cancellation characterization of zero division|that is]], $u$ is not a right- (resp., left-) [[zero-divisor]] > > - The inverse of a two-sided unit is unique > > - Two-sided units form a [[group]] under multiplication (obvious). > > > ---- #### ---- #### 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 > ``` ---- ---- #### ---- #### 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 > ```