Welcome to our community

Be a part of something great, join today!

Affine Varieties - the x-axis in R^2


Well-known member
MHB Site Helper
Jun 22, 2012
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>