1. ## Ring Isomorphism

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?

2. 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?
1)First prove $\displaystyle I\times J\triangleleft R\times S$
2)Define $\displaystyle \phi : (R\times S)/(I\times J) \to (R/I)\times (S/J)$ by $\displaystyle \phi ((a,b)(I\times J)) = aI\times bJ$
3)Prove $\displaystyle \phi$ is well-defined.
4)Prove $\displaystyle \phi$ is one-to-one.
5)Prove $\displaystyle \phi$ is a ring homorphism.

3. Aren't $\displaystyle aI$ and $\displaystyle bJ$ always equal to zero in $\displaystyle R/I$ and $\displaystyle S/J$.
The function maps everything to (0 , 0) = 0?