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 $\displaystyle \mathbb{R}^2 $ is irreducible since it has coordinate ring

$\displaystyle \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

$\displaystyle \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