# generated subring

###### Definition 1

Let $M$ be a nonempty subset of a ring $A$. The intersection^{} of all subrings of $A$ that include $M$ is the smallest subring of $A$ that includes $M$. It is called the subring generated by $M$ and is denoted by $\mathrm{\u27e8}M\mathrm{\u27e9}$.

The subring generated by $M$ is formed by finite sums of monomials of the form :

$${a}_{1}{a}_{2}\mathrm{\cdots}{a}_{n},\text{where}{a}_{1},\mathrm{\dots},{a}_{n}\in M.$$ |

Of particular interest is the subring generated by a family of subrings $E=\{{A}_{i}|i\in I\}$. It is the ring $R$ formed by finite sums of monomials of the form:

$${a}_{{i}_{1}}{a}_{{i}_{2}}\mathrm{\dots}{a}_{{i}_{n}},\text{where}{a}_{{i}_{k}}\in {A}_{{i}_{k}}.$$ |

If $A,B$ are rings, the subring generated by $A\cup B$ is also denoted by $AB$.

In the case when ${A}_{i}$ are fields included in a larger field $A$ then the set of all quotients^{} of elements of $R$ ( the quotient field of $R$) is the composite field ${\bigvee}_{i\in I}{A}_{i}$ of the family $E$. In other words, it is the subfield generated by ${\bigcup}_{i\in I}{A}_{i}$. The notation ${\bigvee}_{i\in I}{A}_{i}$ comes from the fact that the family of all subfields of a field forms a complete lattice^{}.

The of fields is defined only when the respective fields are all included in a larger field.

Title | generated subring |
---|---|

Canonical name | GeneratedSubring |

Date of creation | 2013-03-22 16:57:27 |

Last modified on | 2013-03-22 16:57:27 |

Owner | polarbear (3475) |

Last modified by | polarbear (3475) |

Numerical id | 9 |

Author | polarbear (3475) |

Entry type | Definition |

Classification | msc 20-00 |

Classification | msc 13-00 |

Classification | msc 16-00 |

Related topic | RingAdjunction |

Defines | subring generated by |

Defines | monomials in rings |