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 is irreducible since it has coordinate ring

which is an integral domain."

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

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

Re: Affine Varieties - the x-axis in R^2

It is precisely the first isomorphism for rings.

Define the map: , where and

(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.