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

fischer

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

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

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

can someone help me out?
thanks in advance.

fischer

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

here, i attached the solution.

Attached Files

essay_on_quiz.pdf (35.6 KB, 92 views)

siyacar

Hi,
Is this solution correct?