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Infinite ring with exactly two non trivial maximal ideals

  1. Dec 17, 2012 #1
    Hey!

    Is there an infinite ring with exactly two maximal ideals.

    Thanks in advance
    LiKeMath
     
  2. jcsd
  3. Dec 17, 2012 #2

    micromass

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    What about [itex]\mathbb{R}\times\mathbb{R}[/itex]?
     
  4. Dec 17, 2012 #3

    Erland

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    How is multiplication defined here?
     
  5. Dec 17, 2012 #4

    micromass

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    Pointswise: [itex](a,b)\cdot (c,d)=(ac,bd)[/itex].
     
  6. Dec 19, 2012 #5

    mathwonk

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    i didn't check this, but i would try to take a ring and remove a lot of ideals by inverting elements. e.g. take the integers and look at all rational numbers that do not have factors of 2 or 3 in the bottom. then presumably the only maximal ideals left are (2) and (3).

    another similar construction, in the ring of all continuous functions on [0,1], invert those that do not vanish at either 0 or 1. Then presumably the only maximal ideals left are those that vanish at one of those points.

    I guess this also resembles micromass's example. I.e. take all continuous functions on the 2 point set {0,1} and then you have as maximal ideals the functions that vanish at 0, namely (0,t) and those that vanish at 1, namely (t,0).
     
    Last edited: Dec 19, 2012
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