Originally Posted by

**vincisonfire** Let R and S be rings and let I ▹R, J ▹S be ideals. Prove that

$\displaystyle (R\times S)/(I\times J) \simeq (R/I) \times (S/J) $

I think

$\displaystyle (R\times S)/(I\times J) $ is of the form $\displaystyle (a,b) + (I,J) : a \in R, b \in S $ that is also of the form $\displaystyle (a + I, b + J) : a \in R, b \in S $ because of definition of addition.

This is the "cross set" $\displaystyle (R/I) \times (S/J) $.

Something tells me it can't be that simple. Can you tell me why?