# Infinite ring with exactly two non trivial maximal ideals

1. Dec 17, 2012

### LikeMath

Hey!

Is there an infinite ring with exactly two maximal ideals.

LiKeMath

2. Dec 17, 2012

### micromass

Staff Emeritus
What about $\mathbb{R}\times\mathbb{R}$?

3. Dec 17, 2012

### Erland

How is multiplication defined here?

4. Dec 17, 2012

### micromass

Staff Emeritus
Pointswise: $(a,b)\cdot (c,d)=(ac,bd)$.

5. Dec 19, 2012

### mathwonk

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