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Definition: Irreducible, Prime

Let \((R, +,\cdot)\) be an integral domain. An element $p\in R$ is called a prime element of $R$, if and only if

• $p\neq 0,$ and
• $p$ is not a unit, and
• if $p\mid ab$, then $p\mid a\vee p\mid b,$ i.e. if $p$ divides a product of two elements $a,b\in R,$ then it equals on of them.

An element $p\in R$ is called irreducible, if and only if

• $p\neq 0,$ and
• $p$ is not a unit, and
• if $p\mid ab$, then either $a$ or $b$ is a unit.

Mentioned in:

Definitions: 1 2
Proofs: 3 4
Propositions: 5 6

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References

Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013
2. Koch, H.; Pieper, H.: "Zahlentheorie - Ausgewählte Methoden und Ergebnisse", Studienbücherei, 1976