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.