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I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the primary decompositions of $I$ and $J$?

More precisely, when is true that $Ass(R/I) = Ass(R/J)$? Or at least one is contained in the another one?

For example, let be $I=(t^{3}+ut^{2}, t^{4})$ and $J=(t^{2})$, we have that $I \subset J$, and in this case $Ass(R/J)=\lbrace (t) \rbrace \subset Ass(R/I)=\lbrace (t), (t,u) \rbrace$.

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  • $\begingroup$ $R$ is a formal power series ring over what? A field? A ring? $\endgroup$
    – Mohan
    Oct 16, 2015 at 22:34
  • $\begingroup$ $t^3+ut^2$ is in $I$ but not $J$ in your example, unless I've misunderstood the question. $\endgroup$
    – eric
    Oct 17, 2015 at 14:02
  • $\begingroup$ I'm sorry for text typing errors. R is a formal power series ring over the field of complex numbers, and the ideal $I=(t^{3})$ I had misspelled is actually $I=(t^{2})$. $\endgroup$
    – Otoniel Silva
    Oct 19, 2015 at 13:37
  • $\begingroup$ The question is: what is the relationship of the primes associated with $I$ and $J$ when $I \subset J$? $\endgroup$
    – Otoniel Silva
    Oct 19, 2015 at 13:39

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