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Abstract Algebra

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Integral Domain: Distinction between Primes and Irreducibles

ABSALG-CM4FFR

Let $R$ be a commutative ring with identity $1_R$.

Assume throughout the question that $R$ is an Integral Domain.

Which of the following are always true?

Select ALL that apply.

A

Every irreducible in $R$ is also prime in $R$.​

B

Every prime in $R$ is also irreducible in $R$.

C

The set of primes equals the set of irreducibles in every Principal Ideal Domain (PID) $R$.

D

There are Unique Factorization Domains (UFDs) $R$ whose set of primes does not equal its set of irreducibles.