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Math Help - Affine Varieties - the x-axis in R^2

  1. #1
    Super Member Bernhard's Avatar
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    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:

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

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

    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
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  2. #2
    Member Haven's Avatar
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    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.
    Thanks from Bernhard
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