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.
