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

1. Sep 27, 2008

### fischer

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?

Last edited: Sep 27, 2008
2. Sep 29, 2008

### fischer

never mind. i got it.
the proposition is false....

here, i attached the solution.

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3. Oct 22, 2009

### siyacar

Hi,
Is this solution correct?