# zero divisor

Let $a$ be a nonzero element of a ring $R$.

The element $a$ is a left zero divisor if there exists a nonzero element $b\in R$ such that $a\cdot b=0$. Similarly, $a$ is a right zero divisor if there exists a nonzero element $c\in R$ such that $c\cdot a=0$.

The element $a$ is said to be a zero divisor if it is both a left and right zero divisor. A nonzero element $a\in R$ is said to be a regular element^{} if it is neither a left nor a right zero divisor.

Example: Let $R={\mathbb{Z}}_{6}$. Then the elements $2$ and $3$ are zero divisors, since $2\cdot 3\equiv 6\equiv 0\phantom{\rule{veryverythickmathspace}{0ex}}(mod6)$.

Title | zero divisor |

Canonical name | ZeroDivisor |

Date of creation | 2013-03-22 12:49:59 |

Last modified on | 2013-03-22 12:49:59 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 9 |

Author | cvalente (11260) |

Entry type | Definition |

Classification | msc 13G05 |

Related topic | CancellationRing |

Related topic | IntegralDomain |

Related topic | Unity |

Defines | left zero divisor |

Defines | right zero divisor |

Defines | regular element |