Properties:: [[GCD is a linear combination]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- > [!definition] Definition. ([[greatest common divisor]]) > Let $R$ be a (usually [[commutative ring|commutative]]) [[integral domain]], and let $a,b \in R$. An element $d \in R$ is a **greatest common divisor (gcd)** of $a$ and $b$ if $\langle a,b \rangle \subset \langle d \rangle$ and $\langle d \rangle$ is the smallest [[principal ideal]] of $R$ with this property. > > In other words, $d$ is a gcd of $a$ and $b$ if $d |a$, $d |b$, and $c |a, c | b \implies c | d.$ > We write $d=\text{gcd}(a,b)$. Here, $\langle a,b \rangle$ denotes the [[ideal generated by a subset|ideal generated by]] $a$ and $b$. ^1ba1c8 > [!note] Remark. > Note that greatest common divisors are not defined uniquely by this description; only up to [[divides|associate]] class. Of course, language is often (harmlessly) abused. For example, in $\mathbb{Z}$ there is a convenient way to choose a distinguished element in each class of [[divides|associate]] integers (the nonnegative one). Indeed, this is baked into the specialized definition below... ^note > [!specialization] > The **greatest common [[divides|divisor]]** of $a,b \in \zz \cut \{ 0 \}$ is the largest $d \in \zz$ for which $d | a$ and $d | b$. ^specialization ---- #### ---- #### 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 > ```