# Klein 4-ring

One of the two smallest non-commutative rings is the
Klein 4-ring $(R,+,\cdot )$ where $(R,+)$ is the Klein 4-group $\{0,a,b,c\}$ with $0$ the neutral element and the binary operation^{} “$\cdot $” given by the table

$$\begin{array}{ccccc}\hfill \cdot \hfill & \hfill 0\hfill & \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill \mathrm{\hspace{0.33em}0}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a\hfill & \hfill 0\hfill & \hfill a\hfill & \hfill 0\hfill & \hfill a\hfill \\ \hfill b\hfill & \hfill 0\hfill & \hfill b\hfill & \hfill 0\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill 0\hfill & \hfill c\hfill & \hfill 0\hfill & \hfill c\hfill \end{array}$$ |

Note that this ring has two different right unities $a$ and $c$.

The Klein 4-ring has the subrings $\{0,a\}$, $\{0,b\}$ and $\{0,c\}$ and the two-sided ideal^{} $\{0,b\}$.

Title | Klein 4-ring |

Canonical name | Klein4ring |

Date of creation | 2015-06-04 16:11:18 |

Last modified on | 2015-06-04 16:11:18 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 16 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 20-00 |

Classification | msc 16B99 |

Synonym | Klein’s four-ring |

Synonym | Klein four-ring |

Related topic | Klein4Group |

Related topic | InversesInRings |

Related topic | NonCommutativeRingsOfOrderFour |

Related topic | GroupsInField |

Related topic | Subcommutative |