202507061607

Tags : Ring Theory

Prime and Irreducible elements

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Definition

• An element $a\in R$ is called prime if the ideal $(a)$ is prime.

$$a|bc⟹a|b\text{ or }a|c$$

• An element $a$ is irreducible if $a$ is not a unit and

Some ways to think about irreducible elements:

• $a=bc$ implies $a$ is an associate of $b$ or $c$.
• $a=bc$ implies $(a)=(b)$ or $(a)=(c)$.
• $(a)\subseteq (b)$ implies $(a)=(b)$ or $(b)=(1)$
• $(a)$ is maximal among proper Principle Ideals.

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References

• Prime Ideals and Maximal Ideals
• Principle Ideals