----- > [!proposition] Proposition. ([[unit characterization of associates in commutative integral domains]]) > Let $R$ be a [[commutative ring|commutative]] [[integral domain]]. Then $a,b \in R$ are [[divides|associates]] if and only if $a=ub$ for $u$ a [[unit]] of $R$. ^proposition > [!basicexample] > Put $R=\mathbb{Z}$. The [[unit|units]] of $\mathbb{Z}$ are $1$ and $-1$; this proposition implies that the associates of any $k \in \mathbb{Z}$ are precisely $k$ and $-k$. ^basic-example > [!proof]- Proof. ([[unit characterization of associates in commutative integral domains]]) > > $\to$. Suppose $a,b$ are associates; write $b=ac$ and $a=bd$ for some $c,d \in R$. Then $b=bdc$; the [[cancellation characterization of zero division]] together with the fact that $R$ is an [[integral domain]] implies $1=dc$; therefore $d=u$ is a [[unit]], as required ([[commutative ring|commutativity]] being liberally employed throughout this argument). > $\leftarrow$. Conversely suppose $a=ub$ for $u$ a [[unit]]. Then $b=u ^{-1} a$. So clearly $a,b$ are associates. ----- #### ---- #### 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 > ```