---- > [!definition] Definition. ([[divides]]) > Let $a,b \in R$, where $R$ is a [[commutative ring|commutative]] [[ring]] (usually $\mathbb{Z}$). We say **$a$ divides $b$** if $b \in \langle a \rangle$ — i.e., if there exists $q \in R$ such that $b=aq.$ > > We call $a$ a **divisor** or **factor** of $b$, and say that $b$ **is divisible by** $a$ or **is a multiple of** $a$. We write $a | b$. > >If $a | b$ and $b | a$, (that is, if $\langle a \rangle=\langle b \rangle$) then $a,b$ are called **associates**. > > Here, $\langle a \rangle$ denotes the [[principal ideal|principal]] [[ideal]] of $R$ generated by $a$. > > [!equivalence] > - [[unit characterization of associates in commutative integral domains]] ^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 > ```