ring adjunction
Let be a commutative ring and an extension ring of it. If and commutes with all elements of , then the smallest subring of containing and is denoted by . We say that is obtained from by adjoining to via ring adjunction.
By the about “evaluationhomomorphism”,
where is the polynomial ring in one indeterminate over. Therefore, consists of all expressions whichcan be formed of and elements of the ring by usingadditions, subtractions and multiplications.
Examples: The polynomial rings , the ring of the Gaussian integers, the ring 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 |