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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.

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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.

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  • $\begingroup$ Right, I am aware of CRT but I was curious if one could speed up the decomposition by adding variables (i.e consider some quotient of R[t] ) or some other method. $\endgroup$ – user161954 Sep 25 '14 at 16:20
  • $\begingroup$ Ah, that I don't know $\endgroup$ – Omnomnomnom Sep 25 '14 at 16:23
  • $\begingroup$ Or say when I and J are not coprime, one needs something more than R/K, so one must add variables. I should probably have asked how to do this when I and J are not coprime and when one is allowed to add variables. $\endgroup$ – user161954 Sep 25 '14 at 16:39

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