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Ideals for RxR

  • Thread starter Rederick
  • Start date
  • #1
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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?
 

Answers and Replies

  • #2
12
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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?
 
  • #3
gb7nash
Homework Helper
805
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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
 

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