translation automorphism of a polynomial ring

Let R be a commutative ring, let R[X] be the polynomial ring over R, and let a be an element of R. Then we can define a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath τa of R[X] by constructing the evaluation homomorphism from R[X] to R[X] taking rR to itself and taking X to X+a.

To see that τa is an automorphismMathworldPlanetmath, observe that τ-aτa is the identityPlanetmathPlanetmath on RR[X] and takes X to X, so by the uniqueness of the evaluation homomorphism, τ-aτa is the identity.

Title translationPlanetmathPlanetmath automorphism of a polynomial ring
Canonical name TranslationAutomorphismOfAPolynomialRing
Date of creation 2013-03-22 14:16:13
Last modified on 2013-03-22 14:16:13
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 4
Author archibal (4430)
Entry type Example
Classification msc 12E05
Classification msc 11C08
Classification msc 13P05
Related topic IsomorphismSwappingZeroAndUnity