Associates Irreducibles Primes

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Definition (Associates Irreducibles Primes)

Let $D$ be an integral domain. All elements below are elements of $D$.

• We say that two elements $a$ and $b$ are associates if $a=ub$ for some unit $u$.
• If $a$ is nonzero and not a unit, then we say $a$ is an irreducible if whenever $a=bc$, then either $b$ or $c$ is a unit.
• If $a$ is nonzero and not a unit, then we say $a$ is a prime if $a\mid bc$ implies $a\mid b$ or $a\mid c$.

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