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Def/Irreducible element

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Definition of Irreducible: Suppose that $R$ is an integral domain, such as $\ZZ$ or $\RR[X]$. Suppose that $x \in R$. We say that $x$ is irreducible if the following conditions hold:

  1. $x$ is nonzero.
  2. $x$ is not a unit.
  3. If $y$ divides $x$, then $y$ is a unit or $y$ is a unit multiple of $x$ (i.e., an associate of $x$).

Logical Connections

This definition logically relies on the following definitions and statements: Def/Integral domain, Def/Unit, Def/Divides, Def/Associate

The following statements and definitions logically rely on the material of this page: Def/Factorization into irreducibles, and State/Irreducible implies prime in a PID

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Factorization in rings


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