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Infinite ring with exactly two non trivial maximal ideals 
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#1
Dec1712, 10:50 AM

P: 62

Hey!
Is there an infinite ring with exactly two maximal ideals. Thanks in advance LiKeMath 


#3
Dec1712, 03:31 PM

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#4
Dec1712, 03:32 PM

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Infinite ring with exactly two non trivial maximal ideals



#5
Dec1912, 11:29 AM

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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). 


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