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Here is a nice fact: given a map of unital commutative rings, the extension operation
smallest ideal of containing
combines with the usual preimage to give a monotone Galois connection between the ideal posets of and . That is, the operations form an adjoint pair of functors between these two poset categories, and one has e.g. .
Question: is there any conceptual reason to expect this works? If is a flat ring map, we can describe as , the extension of scalars of the -module . But not in general. Also, is definitely not the restriction of scalars to of the -module . So that is not where this adjunction comes from.
Other question: Is it always true that for substructures which aren't preserved by maps (maybe because they care about the ambient structure too much in some sense), one can "fix" usual the set-theoretic direct image/preimage Galois connection, by taking the generated substructure on the target side?
Edit: Of course you can do this for any substructures preserved by preimage, assuming you have a "substructure generated" in the sense of smallest substructure containing. Indeed, is tautological, and the reverse implication holds because is a substructure containing the set-theoretic image .
You might enjoy thinking about how you can recover this Galois connection from the tensor-hom adjunction between R-modules and S-modules.
I probably can't help with this question, so I'll just ask another question. What's an example of a commutative ring homomorphism and an ideal such that the smallest ideal of containing is not ?
Your question is a little ill formed John. The module isn't generally an ideal of , in that the canonical map isn't injective. This map being injective for all is exactly what it means for to be flat over . Note however that the image of is always the extension of (the bilinear map it corresponds to is multiplication, so the image is the set of sums of products of stuff in and stuff in , which is the extension of ).
An example of a non flat morphism of rings is the projection from to . If we let then is nonzero but the map is zero (this latter bit is easy, just note acts as zero on ). To see observe and so (or less directly, is a nonzero finitely generated module over a local ring and hence by Nakayama's lemma the tensor with the residue field is nonzero)
This helps, thanks a lot
Brendan Murphy said: