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Suppose $I$ and $J$ are two ideals in a polynomial ring $R=\mathbb{Q}[x_1,\dots,x_n]$, what's the relation between the primary decomposition of $I$ in the quotient ring $R/J$ and the primary decomposition of $I+J$ in $R$? In particular it can be assumed $J$ is prime.

The question is raised because all the algorithms for the computation of primary decomposition assume a polynomial ring.

Thanks.

l'etranger
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$I+J=∩_i Q_i$ primary decomposition in $R$ ⇒ $(I+J)/J=∩_i (Q_i /J)$ primary decomposition in $R/J$, and viceversa.