- #1

- 62

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Hey!

Is there an infinite ring with exactly two maximal ideals.

Thanks in advance

LiKeMath

Is there an infinite ring with exactly two maximal ideals.

Thanks in advance

LiKeMath

- Thread starter LikeMath
- Start date

- #1

- 62

- 0

Hey!

Is there an infinite ring with exactly two maximal ideals.

Thanks in advance

LiKeMath

Is there an infinite ring with exactly two maximal ideals.

Thanks in advance

LiKeMath

- #2

- 22,089

- 3,286

What about [itex]\mathbb{R}\times\mathbb{R}[/itex]?

- #3

Erland

Science Advisor

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How is multiplication defined here?What about [itex]\mathbb{R}\times\mathbb{R}[/itex]?

- #4

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Pointswise: [itex](a,b)\cdot (c,d)=(ac,bd)[/itex].How is multiplication defined here?

- #5

mathwonk

Science Advisor

Homework Helper

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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).

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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