What are the properties of ideals in a commutative ring with identity 1?

[wegmanstuna]

Alright, I need some help with this problem (mainly just to get started):
Let A and B be ideals of R, such that:
A+B={a+b / a in A ,b in B}
AB={aibi / ai in A , bi in B}
A:B={x in R / xb in A} are all ideals of R
Show that A+B, AB, and A:B are ALL the ideals of R, where R is a commutative ring with identity 1.


Thanks guys

 

[morphism]

This doesn't make sense. There's no reason to believe that those things will be all the ideals of R. In fact, in general they won't be. Are you sure the problem isn't to simply verify that A+B, AB and A:B are ideals? In which case there's still a problem: what you wrote down for "AB" isn't good - it won't be an ideal in general. Should there be a summation sign before aibi?

 

[wegmanstuna]

Yes you are right there should be a summation sign before aibi, that was my bad. But I have been working with someone else on this problem and we came to the conclusion that, your right, they cannot possibly represent all the ideals of R. But is it possible that they represent all the properties of ideals in general? I think that may have been the problem; to show that they represent all the properties of ideals.

 

[Pere Callahan]

What do you mean by "they represent all the properties of ideals".

If they have all the properties of ideals then they are ideals and that is what morphism supposed in his reply...that the taks might be to prove that A+B etc are ideals.
This does not mean of course that there are no other ideals...