---- > [!definition] Definition. ([[prime element of an integral domain]]) > Let $R$ be an [[integral domain]]. An element $a \in R$ is **prime** if the [[principal ideal|principal]] [[ideal]] $\langle a \rangle$ is a [[prime ideal]]. ^definition > [!equivalence] > It is fast that $a \in R$ is prime if and only if $a$ is not a [[unit]] of $R$ and $a | bc \implies (a | b \text{ or } a | c).$ > Thus the specialization below makes sense. ^equivalence [[prime iff maximal iff irreducible for nonzero ideals in a PID]] > [!specialization] > - [[Euclid's Lemma]] is the assertion that [[prime number|prime numbers]] are prime elements in the [[ring]] $(\mathbb{Z}, +, \cdot)$. ^specialization > [!proof]+ Proof of Equivalence. > First, note that $\begin{align} a \text{ is a unit } \iff& 1 \in \langle a \rangle \\ \iff & \langle a \rangle =R \\ \iff& \langle a \rangle =1 \end{align}$ > and hence the declaration "$a$ is not a unit" aligns with the assumption that $I \neq \langle 1 \rangle$ in the definition of [[prime ideal]]. > Moreover, from [[prime ideal#^equivalence|the equivalence]] in [[prime ideal]] we get $\begin{align} R / \langle a \rangle, \langle a \rangle \neq \langle 1 \rangle, \text{ is an integral domain } \iff& (bc \in \langle a \rangle \implies b \in \langle a \rangle \text{ or } c \in \langle a \rangle) \\ \iff & a |bc \implies (a | b \text{ or } a |c ); \end{align}$ this is the equivalence claimed. ^proof ---- #### ---- #### 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 > ```