Finding Ideals in RxR: A Complete Guide

[Rederick]

Homework Statement

Let S be a ring = RxR (real#,real#). Find all the ideals in RxR.

Homework Equations

We were told that there are only 4.

The Attempt at a Solution

I can only think of these 4 sub-rings of S, (R,0), (0,R), (R,R) and (0,0). And each seem to be ideal. Are these the correct 4? What other sub-rings of RxR are there but not ideal?

 

[Rederick]

Thinking more about it,T={(100a,100b)|a,b are elements in R} is a sub-ring but not ideal since (100a,100b)(1/2,1/2)=(50a,50b) which is not in T.

How would I say that in general to show that there are only 4?

 

[gb7nash]

I can only think of these 4 sub-rings of S, (R,0), (0,R), (R,R) and (0,0). And each seem to be ideal. Are these the correct 4?

This looks fine to me. Just remember one condition of a set being ideal is that the set must be a subgroup of RXR under addition. You can check the properties for (R,0), (0,R), (R,R) and (0,0)

Looks

What other sub-rings of RxR are there but not ideal?

ZXZ