Ring theory MOC

Prime element

Let 𝑅 be a ring. A nonzero element 𝜋 ∈𝑅 is prime iff it is not a unit and it satisfies Euclid’s lemma: ring Whenever 𝜋 ∣𝑥𝑦 for 𝑥,𝑦 ∈𝑅 then 𝜋 ∣𝑥 or 𝜋 ∣𝑦.

(∀𝑥,𝑦∈𝑅)[𝜋∣𝑥𝑦⟹[𝜋∣𝑥]∨[𝜋∣𝑦]]

This is one way to generalize the Prime number to an arbitrary ring.¹

Properties

• All primes are irreducible in an integral domain

See also

• Prime ideal

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Footnotes

1. 2022. Algebraic number theory course notes, §1.1, p. 1 ↩