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