Prime element of a ring

An element of an integral domain is prime if it has the property that \(p \mid ab\) implies \(p \mid a\) or \(p \mid b\). Equivalently, if its generated ideal is prime in the sense that \(ab \in \langle p \rangle\) implies either \(a\) or \(b\) is in \(\langle p \rangle\).

Be aware that “prime” in ring theory does not correspond exactly to “prime” in number theory (the correct abstraction of which is irreducibility). It is the case that they are the same concept in the ring \(\mathbb{Z}\) of integers (proof), but this is a nontrivial property that turns out to be equivalent to the fundamental theorem of arithmetic (proof).



- Primes are always irreducible; a proof of this fact appears on the page on irreducibility, along with counterexamples to the converse.