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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!

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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.

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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

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