Prime element

In ring theory an element $p$ of an integral domain $R$ is said to be prime if:

  • $p$ is not a unit.
  • If $p|ab$ for any $a,b\in R$ then $p|a$ or $p|b$.

Equivalently, we can say that $p$ is prime iff $(p)$ is a prime ideal in $R$.

Any prime element $p\in R$ is clearly irreducible in $R$. (Indeed if $p=ab$, then we would have $p|ab$, so $p$ would have to divide one of $a$ and $b$, WLOG $a$. Then $a=pc$ for some $c\in R$, so $p = ab = pbc$, so $bc=1$, and hence $b$ is a unit.) The converse of this holds in any unique factorization domain, but it does not hold in a general integral domain.

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