# ring adjunction

Let $R$ be a commutative ring and $E$ an extension ring of it. If $\alpha \in E$ and commutes with all elements of $R$, then the smallest subring of $E$ containing $R$ and $\alpha $ is denoted by $R[\alpha ]$. We say that $R[\alpha ]$ is obtained from $R$ by adjoining $\alpha $ to $R$ via ring adjunction.

By the about “evaluation homomorphism”,

$$R[\alpha ]=\{f(\alpha )\mid f(X)\in R[X]\},$$ |

where $R[X]$ is the polynomial ring in one indeterminate over $R$. Therefore, $R[\alpha ]$ consists of all expressions which can be formed of $\alpha $ and elements of the ring $R$ by using additions, subtractions and multiplications.

Examples: The polynomial rings $R[X]$, the ring $\mathbb{Z}[i]$ of the Gaussian integers^{}, the ring $\mathbb{Z}[\frac{-1+i\sqrt{3}}{2}]$ of Eisenstein integers^{}.

Title | ring adjunction |

Canonical name | RingAdjunction |

Date of creation | 2014-02-18 14:13:46 |

Last modified on | 2014-02-18 14:13:46 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 17 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 13B25 |

Classification | msc 13B02 |

Related topic | GeneratedSubring |

Related topic | FiniteRingHasNoProperOverrings |

Related topic | GroundFieldsAndRings |

Related topic | PolynomialRingOverIntegralDomain |

Related topic | AConditionOfAlgebraicExtension |

Related topic | IntegralClosureIsRing |