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Infinite ring with exactly two non trivial maximal ideals

by LikeMath
Tags: ideals, infinite, maximal, ring, trivial
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LikeMath
#1
Dec17-12, 10:50 AM
P: 62
Hey!

Is there an infinite ring with exactly two maximal ideals.

Thanks in advance
LiKeMath
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micromass
#2
Dec17-12, 01:03 PM
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What about [itex]\mathbb{R}\times\mathbb{R}[/itex]?
Erland
#3
Dec17-12, 03:31 PM
P: 345
Quote Quote by micromass View Post
What about [itex]\mathbb{R}\times\mathbb{R}[/itex]?
How is multiplication defined here?

micromass
#4
Dec17-12, 03:32 PM
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Infinite ring with exactly two non trivial maximal ideals

Quote Quote by Erland View Post
How is multiplication defined here?
Pointswise: [itex](a,b)\cdot (c,d)=(ac,bd)[/itex].
mathwonk
#5
Dec19-12, 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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