---- > [!definition] Definition. ([[PID]]) > An [[integral domain]] $R$ is a **PID (Principal Ideal Domain)** if every [[ideal]] of the [[ring]] $R$ is a [[principal ideal]]. ^definition > [!basicexample] > - $(\mathbb{Z}, +, \cdot)$ is a PID. Indeed, every [[ideal]] $\mathbb{Z}$ looks like $I=n\mathbb{Z}$ for some $n \in \mathbb{N}$ (for these are the [[subgroup|subgroups]] of $\mathbb{Z}$ and by definition as 'multiples' they absorb multiplication). Since $n\mathbb{Z}=\langle n \rangle$, this shows $I$ is principal. This explains 'why' the [[greatest common divisor]] behaves as it does in $\mathbb{Z}$: if $m,n \in \mathbb{Z}$ then the [[ideal generated by a subset|ideal]] $\langle m,n \rangle$ must be [[principal ideal|principal]], thus $\langle m,n \rangle=\langle d \rangle$ for some integer $d$... manifestly, $d=\text{gcd}(m,n)$. Maybe this sheds some intuition on [[GCD is a linear combination]]. ^basic-example > [!basicnonexample] > The [[polynomial 4|polynomial ring]] $\mathbb{Z}[x]$ is not a PID. Indeed, the [[ideal]] $\langle 2,x \rangle$ cannot be [[ideal generated by a subset|generated by]] a single element. ^nonexample ---- #### ---- #### 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 > ```