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.