# non-commutative rings of order four

Up to isomorphism^{}, there are two non-commutative rings of order (http://planetmath.org/OrderRing) four. Since all cyclic rings are commutative^{} (http://planetmath.org/CommutativeRing), one can immediately deduce that a ring of order four must have an additive group^{} that is isomorphic to ${\mathbb{F}}_{2}\oplus {\mathbb{F}}_{2}$.

One of the two non-commutative rings of order four is the Klein 4-ring, whose multiplication table is given by:

$$\begin{array}{ccccc}\hfill \cdot \hfill & \hfill 0\hfill & \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a\hfill & \hfill 0\hfill & \hfill a\hfill & \hfill 0\hfill & \hfill a\hfill \\ \hfill b\hfill & \hfill 0\hfill & \hfill b\hfill & \hfill 0\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill 0\hfill & \hfill c\hfill & \hfill 0\hfill & \hfill c\hfill \end{array}$$ |

The other is closely related to the Klein 4-ring. In fact, it is anti-isomorphic to the Klein 4-ring; that is, its multiplication table is obtained by swapping the of the multiplication table for the Klein 4-ring:

$$\begin{array}{ccccc}\hfill \cdot \hfill & \hfill 0\hfill & \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a\hfill & \hfill 0\hfill & \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill b\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill c\hfill & \hfill 0\hfill & \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \end{array}$$ |

Title | non-commutative rings of order four |
---|---|

Canonical name | NoncommutativeRingsOfOrderFour |

Date of creation | 2013-03-22 17:09:24 |

Last modified on | 2013-03-22 17:09:24 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 12 |

Author | Wkbj79 (1863) |

Entry type | Topic |

Classification | msc 20-00 |

Classification | msc 16B99 |

Related topic | Klein4Ring |

Related topic | OppositeRing |

Related topic | ExampleOfKlein4Ring |