# 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