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Homering adjunction

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# 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 Theorem 1 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.

## Mathematics Subject Classification

13B25*no label found*13B02

*no label found*

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