# 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).

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