ring adjunction

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

By the about “evaluation homomorphism”,


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

Examples:  The polynomial rings R[X], the ring [i] of the Gaussian integersMathworldPlanetmath, the ring [-1+i32] of Eisenstein integersMathworldPlanetmath.

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