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

• November 8th 2013, 12:30 AM
Bernhard
Affine Varieties - the x-axis in R^2
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
• November 26th 2013, 08:48 PM
Haven
Re: Affine Varieties - the x-axis in R^2
It is precisely the first isomorphism for rings.

Define the map: $R[x,y] \rightarrow R[x,y]$, where $x\mapsto x$ and $y\mapsto 0$
(Notice the definition of the map coincides with what we do when we take a quotient).
Find the kernel and image of this map and you're golden.