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Sum of two prime ideals is prime ?

  1. Mar 29, 2008 #1
    i was looking for a counter example.

    and, i've not been able to think of any.
     
  2. jcsd
  3. Mar 29, 2008 #2
    look at Z
     
  4. Mar 29, 2008 #3

    mathwonk

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    the sum of two ideals is the ideal generated by their union right?

    thus geometrically it is the ideal of the intersection of the two zero loci.

    so look for a pair of irreducible algebraic sets whose intersection is reducible,

    (like a quadric surface and a tangent plane.)

    i.e. a prime ideal is one that has a (reduced and) irreducible zero set.
     
  5. Mar 30, 2008 #4
    i did not understand the geometric bit.
    well i considered the principal ideals <2> and <3>
    their union includes 1 which is a unit in Z, so the ideal of the sum is nothing but Z itself right ?
    and that can't be prime by definition ? (since an ideal P is prime => P /= R (the ring in consideration))
    more hints please.
     
  6. Mar 30, 2008 #5
    yea that's fine, just notice 1 = -1*2 + 1*3

    another way to think about it is in terms of existence of gcd's. gcd(2, 3) = 1, so there are x, y in Z such that 2x + 3y = 1 and this is in <2> + <3>, but this is an ideal, so x = x*1 is in <2> + <3> for all x in Z, so yea Z = <2> + <3>, probably overkill but a useful observation

    this can be generalized and is really useful, if R is a pid, then Ra + Rb = Rd where d = gcd(a, b)

    when looking for counterexamples always think simple(doesn't always work but sometimes it does), like for example 2Z and 3Z are prime but 2Z n 3Z = 6Z is not, so the intersection of prime ideals isn't necessarily prime
     
    Last edited: Mar 30, 2008
  7. Mar 30, 2008 #6

    Hurkyl

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    Think of rings like R[x, y]. Algebraic curves (like the parabola y - x^2 = 0) correspond to ideals (like the ideal <y - x^2>). Sums of ideals relate to intersections of curves. Can you work out why? Do you see how a non-prime ideal corresponds, in some sense, into a curve that is the union of two or more other curves?
     
  8. Mar 30, 2008 #7

    mathwonk

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    a prime ideal of a cone is (Z^2 -X^2-Y^2).

    a prime ideal of a plane is (Y).

    the sum of these ideals (Z^2 -X^2-Y^2, Y) = (Z^2 -X^2, Y), is the ideal of the intersection, which is the two lines Z=X, Z=-X, in the X,Z plane.
     
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