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Affine Varieties - the x-axis in R^2

  1. Nov 8, 2013 #1
    In Dummit and Foote, Chapter 15, Section 15.2 Radicals and Affine Varieties, Example 2, page 681 begins as follows:

    -----------------------------------------------------------------------------------------
    "The x-axis in [itex] \mathbb{R}^2 [/itex] is irreducible since it has coordinate ring

    [itex] \mathbb{R}[x,y]/(y) \cong \mathbb{R}[x] [/itex]

    which is an integral domain."

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

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

    [itex] \mathbb{R}[x,y]/(y) \cong \mathbb{R}[x] [/itex] 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
     
  2. jcsd
  3. Nov 9, 2013 #2
    Consider ##\Phi: R[X,Y]\rightarrow R[X]: \sum \alpha_{ij}X^i Y^j \rightarrow \sum \alpha_{i0}X^i##. So we evaluate the polynomial in ##0##. These evaluation maps usually work in these contexts.
     
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