# example of Klein 4-ring

The Klein $4$-ring $K$ can be represented by as a left ideal^{} of $2\times 2$-matrices
over the field with two elements ${\mathbb{Z}}_{2}$. Doing so helps to explain
some of the unnatural properties of this nonunital ring and is an example of how
many nonunital rings can often be understood as very natural subobjects of unital
rings.

$$K=\{\left[\begin{array}{cc}\hfill x\hfill & \hfill 0\hfill \\ \hfill y\hfill & \hfill 0\hfill \end{array}\right]:x,y\in {\mathbb{Z}}_{2}\}$$ | (1) |

To match the product with with the table, use

$a$ | $:=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$ | ||

$b$ | $:=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right],$ | ||

$c$ | $:=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right].$ |

Here the properties of the abstract multiplication table of $K$ (given
here (http://planetmath.org/Klein4Ring)) can be
seen as rather natural properties of unital rings. That is, the elements
$a$ and $c$ are idempotents^{} in the ring ${M}_{2}({\mathbb{Z}}_{2})$ and so they
behave similar to identities^{}, and $b$ is nilpotent^{} so that its annihilating
property is expected as well.

The second noncommutative nonunital ring of order 4 is the transpose of these matrices, that is, a right ideal of ${M}_{2}({\mathbb{Z}}_{2})$.

Viewed in this way we recognize the Klein $4$-ring as part of an infinite family of similar nonunital rings of left/right ideals of a unital ring. Some authors prefer to treat such objects only as ideals and not as rings so that the properties are always given the background of a more familiar structure.

Title | example of Klein 4-ring |
---|---|

Canonical name | ExampleOfKlein4ring |

Date of creation | 2013-03-22 17:41:59 |

Last modified on | 2013-03-22 17:41:59 |

Owner | Algeboy (12884) |

Last modified by | Algeboy (12884) |

Numerical id | 9 |

Author | Algeboy (12884) |

Entry type | Example |

Classification | msc 16B99 |

Classification | msc 20-00 |

Related topic | NonCommutativeRingsOfOrderFour |