Affine Varieties - the x-axis in R^2

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In summary, the x-axis in \mathbb{R}^2 is irreducible since its coordinate ring, \mathbb{R}[x,y]/(y), is isomorphic to \mathbb{R}[x], which is an integral domain. This isomorphism is established by applying the First Isomorphism Theorem for rings and using the evaluation map ##\Phi: R[X,Y]\rightarrow R[X]: \sum \alpha_{ij}X^i Y^j \rightarrow \sum \alpha_{i0}X^i##.
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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 [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."

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

1. What is an affine variety?

An affine variety is a geometric object in algebraic geometry that is defined as the set of common zeros of a set of polynomial equations in a finite number of variables. In other words, it is a solution set to a system of polynomial equations.

2. What is the x-axis in R^2?

The x-axis in R^2 is a line on a two-dimensional coordinate plane that extends infinitely in both the positive and negative directions. It is represented by the equation y=0.

3. How is the x-axis related to affine varieties?

When considering affine varieties in R^2, the x-axis is often used as a reference line for determining the solutions to polynomial equations. This is because the x-axis represents the set of points where the y-coordinate is equal to 0, so any solution to a polynomial equation on the x-axis will have a y-value of 0.

4. Can an affine variety intersect the x-axis in R^2 at more than one point?

Yes, an affine variety can intersect the x-axis in R^2 at more than one point. This occurs when the polynomial equations defining the affine variety have more than one solution on the x-axis.

5. How are affine varieties in R^2 graphed on the x-axis?

Affine varieties in R^2 can be graphed on the x-axis by plotting the points that satisfy the polynomial equations defining the variety. This can be done by substituting different x-values into the equations and solving for the corresponding y-values. The resulting points can then be plotted on the x-axis to visualize the affine variety.

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