Ideals for RxR

  • Thread starter Rederick
  • Start date
  • #1
12
0

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
0
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
1
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
 

Related Threads on Ideals for RxR

  • Last Post
Replies
6
Views
1K
Replies
3
Views
293
Replies
1
Views
2K
Replies
5
Views
943
  • Last Post
Replies
9
Views
1K
Replies
1
Views
590
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
1
Views
818
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
Top