Assume $R_1$ and $R_2$ are two commutative rings with identities, their direct product ring is $R_1\times R_2$, and its ideal can be in the form of $I_1\times I_2$. When considering $R_1\times R_2$/$I_1\times I_2$, people use $R_1\times R_2$/$I_1\times I_2 \approxeq R_1/I_1 \times R_2/I_2$. But why we need to use isometry? I think the quotient should be just $R_1/I_1 \times R_2/I_2$ by definition, but I can't find any definition about $R_1\times R_2$/$I_1\times I_2$. So what's the definition of quotient of direct product ring? We can only use the isometric form to define the quotient? Thank you!
The reason we consider "isomorphisms" instead of "equality" is this:
An element of $(R_1 \times R_2)/(I_1 \times I_2)$ is a "coset of a pair":
$(r_1,r_2) + (I_1 \times I_2)$ (Think of $R_1 \times R_2$ as just "a big ring" and $I_1 \times I_2$ a "big ideal").
An element of $R_1/I_1 \times R_2/I_2$ is "a pair of cosets":
$(r_1 + I_1,r_2 + I_2)$, which is not quite the same thing.
define$$\phi : R_1\times R_2 \to R_1/I_1 \times R_2/I_2$$ such that $$\phi (a,b)= (a+I_1 , b+I_2)$$ then $$ker\phi = I_1\times I_2 .$$ Now use First isomorphism theorem