# Writing a direct product of rings over a ring “as one”.

Let $R$ be a commutative ring and suppose that $R/I \oplus R/J$ as ring where I and J are ideals not coprime. Suppose that I wanted to write this as $R[x_1,\ldots ,x_n]/K$ where K is an ideal of the polynomial ring. Are there any economic way of doing this if I say, knew the generators of I and J? Adding variables is OK, if it makes the process more slick.

I edited the question to make it more ask what I wanted to ask. To ease the process, you can assume R is Noetherian.

If $I + J = R$, then we have the following: let $K = I \cdot J$ (the product of the two ideals). Then we have $$R/K \cong R/I \oplus R/J$$ See the Chinese remainder theorem for the reverse of what you're looking for.