# Affine Varieties - the x-axis in R^2

1. Nov 8, 2013

### Math Amateur

In Dummit and Foote, Chapter 15, Section 15.2 Radicals and Affine Varieties, Example 2, page 681 begins as follows:

-----------------------------------------------------------------------------------------
"The x-axis in $\mathbb{R}^2$ is irreducible since it has coordinate ring

$\mathbb{R}[x,y]/(y) \cong \mathbb{R}[x]$

which is an integral domain."

------------------------------------------------------------------------------------------

$\mathbb{R}[x,y]/(y) \cong \mathbb{R}[x]$ is established.

I suspect it comes from applying the First (or Fundamental) Isomorphism Theorem for rings ... but I am unsure of the mappings involved and how they are established

Would appreciate some help>

Peter

2. Nov 9, 2013

### R136a1

Consider $\Phi: R[X,Y]\rightarrow R[X]: \sum \alpha_{ij}X^i Y^j \rightarrow \sum \alpha_{i0}X^i$. So we evaluate the polynomial in $0$. These evaluation maps usually work in these contexts.