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Does Macaulay2 compute contractions of ideals under ring homomorphisms. Specifically, if $R\subseteq S$ is a ring extension (say polynomial rings over $\mathbb{Q}$ which can be specified in M2) and $I$ is an ideal in $S$ given by generators, is there a command to compute $I\cap R$?

EDIT: The eliminate command is supposed to do what I want, except when I use it the output is an ideal in the original ring.

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You could set f=map(S/I,R) and obtain the intersection as ker(f).

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    I've located the following earlier accounts of yours: http://math.stackexchange.com/users/15341, http://math.stackexchange.com/users/15825, http://math.stackexchange.com/users/16790, http://math.stackexchange.com/users/19253, http://math.stackexchange.com/users/20407, http://math.stackexchange.com/users/21246. If you want to merge them into your current registered account, please flag for moderator attention (at the bottom of your answer you have the "flag" link. Click it and explain the situation briefly in the "other" field.)2012-01-10
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More generally there is also the function preimage which takes $f$ a function from $R$ to $S$ and $I$ an ideal in $S$ and outputs $I^c$ in $R$ http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6/share/doc/Macaulay2/Macaulay2Doc/html/_preimage.html