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Abstract Algebra

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Irreducible vs Prime in UFD and PID

ABSALG-L3NWL8

Let $R$ be an integral domain and $a$ be a nonzero non unit of $R$. Recall that:

(i) $a$ is called irreducible if whenever $a=bc$ for some $b, c\in R$, then $b$ or $c$ is a unit.
(ii) $a$ is prime if whenever $a|bc$ for some $b, c\in R$, then $a|b$ or $a|c$.

Note that every prime element is irreducible.

Let $R=\mathbb{Z}[\sqrt{-3}]:=\{a+b\sqrt{-3}:a, b\in \mathbb{Z}\}$.

Which of the following is true about $R$?

A

$2$ is irreducible in $R$.

B

$2$ is prime in $R$.

C

$R$ is a UFD.

D

$R$ is a PID.