Ideals of direct product of rings are direct product of respective ideals?

fischer
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I want to answer this question:
Find all the ideals of the direct product of rings R \times S.
(I think this means show that the ideals are I \times J where I, J are ideals of R, S, respectively.)

I think the problem is that I don't know how to show that any ideal of R \times S is of the form A \times B, where A \subset R, B \subset S. Showing that each are ideals should follow easily enough.

So I made attemps to prove that (a, m), (b, n) \in K iff (a, n), (b, m) \in K (where K is an ideal of R \times S), without success...

can someone help me out?
thanks in advance.
 
Last edited:
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never mind. i got it.
the proposition is false...

here, i attached the solution.
 

Attachments

Hi,
Is this solution correct?
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

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Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

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Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
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