[[Ring theory MOC]]
# Prime element

Let $R$ be a [[ring]].
A nonzero element $\pi \in R$ is **prime** iff it is not a unit and it satisfies [[Euclid's lemma]]: #m/def/ring 
Whenever $\pi \mid xy$ for $x,y \in R$ then $\pi \mid x$ or $\pi \mid y$.
$$
\begin{align*}
(\forall x,y \in R)[\pi \mid xy \implies [\pi \mid x] \lor [\pi \mid y]]
\end{align*}
$$
This is one way to generalize the [[Prime number]] to an arbitrary ring.[^2022]

  [^2022]: 2022\. [[Sources/@bakerAlgebraicNumberTheory2022|Algebraic number theory course notes]], §1.1, p. 1
## Properties

- [[All primes are irreducible in an integral domain]]

## See also

- [[Prime ideal]]

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