Affine Varieties - the x-axis in R^2

[Peter]

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

-----------------------------------------------------------------------------------------
"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."

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

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