# Infinite ring with exactly two non trivial maximal ideals

by LikeMath
Tags: ideals, infinite, maximal, ring, trivial
 P: 62 Hey! Is there an infinite ring with exactly two maximal ideals. Thanks in advance LiKeMath
 Mentor P: 16,593 What about $\mathbb{R}\times\mathbb{R}$?
P: 302
 Quote by micromass What about $\mathbb{R}\times\mathbb{R}$?
How is multiplication defined here?

Mentor
P: 16,593

## Infinite ring with exactly two non trivial maximal ideals

 Quote by Erland How is multiplication defined here?
Pointswise: $(a,b)\cdot (c,d)=(ac,bd)$.
 Sci Advisor HW Helper P: 9,422 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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