It is well known that intersection and sum of polynomial ideals from the same ring are lattice operations. I wonder if this is still true for ideals from different rings (over the same field).

Specifically, let I be an ideal over the ring k[x,y], while J be an ideal over k[y,z]. (I have deliberately chosen the special case of three variables in order to keep notation as simple as possible). Then, the sum of the ideals is defined as:

$ I + J = \{ f(x,y)+g(y,z) | f \in I \wedge g \in J \} $

The theorem that the basis of the sum is concatenation of the bases (which can be used as an alternative definition of the sum) generalizes verbatim.

Intersection of I and J is defined over the ring k[y]. First we eliminate variable x in I, and eliminate z in J, then intersect the elimination ideals over k[y].

Here are the lattice properties:

  • $ I + I = I $ (same as in classic case)
  • $ I + J = J + I $ (easy)
  • $ (I + J) + K = I + (J + K) $ (obvious in alternative definition)
  • $ I \cap I = I $ (same as in classic case)
  • $ I \cap J = J \cap I $ (easy, or redundant anyway)
  • $ (I \cap J) \cap K = I \cap (J \cap K) $ ?
  • $ I + (I \cap J) = I $ ??

Perhaps I can manage to prove associativity of ideal intersection if I establish simpler lemma that elimination of variables commutes with intersection. However, I'm entirely lost trying to prove absorption.

  • $\begingroup$ Hint: How do you define intersections of ideals? $\endgroup$ – NasuSama Feb 11 '14 at 2:20
  • $\begingroup$ Are you implying that $ I \cap (I + J) = I $ easier to prove? ( containment $ I \cap (I + J) \subset I $ is almost immediate). $\endgroup$ – Tegiri Nenashi Feb 11 '14 at 2:27
  • $\begingroup$ Nope, I want you to look at the lattice properties you were unable to prove and determine how to approach such proofs. $\endgroup$ – NasuSama Feb 11 '14 at 2:28

The sum of ideals is the smallest ideal containing a set-theoretic union of ideals (which is a set, not an ideal).

Therefore, we have two lattice set operations together with closure operator on top of one of them, which is well known mechanism of generating new lattice from existing lattice.

Motivation for this question is here.


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