Affine Varieties - the x-axis in R^2

[Math Amateur]

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

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"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."

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Can someone please help me to show formally and rigorously how the isomorphism

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

 

[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.