# unitization

The operation of unitization allows one to add a unity element to an algebra. Because of this construction, one can regard any algebra as a subalgebra of an algebra with unity. If the algebra already has a unity, the operation creates a larger algebra in which the old unity is no longer the unity.

Let $\mathrm{\pi \x9d\x90\x80}$ be an algebra over a ring $\mathrm{\pi \x9d\x90\x91}$ with unity $1$. Then, as a module, the unitization of $\mathrm{\pi \x9d\x90\x80}$ is the direct sum^{} of $\mathrm{\pi \x9d\x90\x91}$ and $\mathrm{\pi \x9d\x90\x80}$:

$${\mathrm{\pi \x9d\x90\x80}}^{+}=\mathrm{\pi \x9d\x90\x91}\beta \x8a\x95\mathrm{\pi \x9d\x90\x80}$$ |

The product operation is defined as follows:

$$(x,a)\beta \x8b\x85(y,b)=(x\beta \x81\u2019y,a\beta \x81\u2019b+x\beta \x81\u2019b+y\beta \x81\u2019a)$$ |

The unity of ${\mathrm{\pi \x9d\x90\x80}}^{+}$ is $(1,0)$.

It is also possible to unitize any ring using this construction if one regards the ring as an algebra over the ring of integers^{} (http://planetmath.org/Integer). (See the entry every ring is an integer algebra for details.) It is worth noting,
however, that the result of unitizing a ring this way will always be a ring whose unity has zero characteristic. If one
has a ring of finite characteristic $k$, one can instead regard it as an algebra over ${\mathrm{\beta \x84\u20ac}}_{k}$ and unitize
accordingly to obtain a ring of characteristic $k$.

The construction described above is often called βminimal unitizationβ. It is in fact minimal, in the sense that every other unitization contains this unitization as a subalgebra.

Title | unitization |
---|---|

Canonical name | Unitization |

Date of creation | 2013-03-22 14:47:36 |

Last modified on | 2013-03-22 14:47:36 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 16-00 |

Classification | msc 13-00 |

Classification | msc 20-00 |

Synonym | minimal unitization |