# prime element is irreducible in integral domain

###### Theorem.

Every prime element^{} of an integral domain^{} is irreducible.

###### Proof.

Let $D$ be an integral domain, and let $a\in D$ be a prime element. Assume $a=bc$ for some $b,c\in D$.

Clearly $a\mid bc$, so since $a$ is prime, $a\mid b$ or $a\mid c$. Without loss of generality, assume $a\mid b$, and say $at=b$ for some $t\in D$.

If $1$ is the unity of $D$, then

$$1b=b=at=(bc)t=b(ct).$$ |

Since $D$ is an integral domain, $b$ can be cancelled, giving $1=ct$, so $c$ is a unit. ∎

Title | prime element is irreducible in integral domain |
---|---|

Canonical name | PrimeElementIsIrreducibleInIntegralDomain |

Date of creation | 2013-03-22 17:15:29 |

Last modified on | 2013-03-22 17:15:29 |

Owner | me_and (17092) |

Last modified by | me_and (17092) |

Numerical id | 7 |

Author | me_and (17092) |

Entry type | Theorem |

Classification | msc 13G05 |

Related topic | IrreducibleOfAUFDIsPrime |