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单词 RingAdjunction
释义

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 “evaluationhomomorphism”,

R[α]={f(α)f(X)R[X]},

where R[X] is the polynomial ring in one indeterminate overR.  Therefore, R[α] consists of all expressions whichcan be formed of α and elements of the ring R by usingadditions, subtractions and multiplications.

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

Titlering adjunction
Canonical nameRingAdjunction
Date of creation2014-02-18 14:13:46
Last modified on2014-02-18 14:13:46
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id17
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 13B25
Classificationmsc 13B02
Related topicGeneratedSubring
Related topicFiniteRingHasNoProperOverrings
Related topicGroundFieldsAndRings
Related topicPolynomialRingOverIntegralDomain
Related topicAConditionOfAlgebraicExtension
Related topicIntegralClosureIsRing
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更新时间:2025/5/4 6:14:24