All primes are irreducible in an integral domain

Let $𝐷$ be an integral domain and $𝜋 \in 𝐷$ be a prime element. Then $𝜋$ is also an irreducible element. ring

Proof

Suppose $𝜋 = 𝑎 𝑏$ with $𝑎, 𝑏 \in 𝑅$ . Then $𝜋 ∣ 𝑎 𝑏$ , so without loss of generality $𝑎 = 𝜋 𝑢$ . Thus $𝜋 1 = 𝑎 𝑏 = 𝜋 𝑢 𝑏$ so by the cancellation property $𝑢 𝑏 = 1$ , whence $𝑏$ is a unit. Therefore $𝜋$ is irreducible.

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All primes are irreducible in an integral domain

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