I have been trying to go through some algebraic geometry using the nonstandard framework. Noetherian rings are of course fundamental in this subject and it is characterized by the attribute that every ideal is finitely generated.
Let $R$ be a ring and $I$ is an ideal in $R$. Let $^*R$ be the enlargment of $R$. There are two entities created in the enlargment.
- $^*I = $ the extension of $I$ in this enlargment.
- $(I) = $ the smallest ideal in $^*R$ containing $I$.
It seems that characterization of Noetherian rings mentioned above equates to saying that $(I) = {}^*I$. The proof of this is are simply applications of the transfer and idealization principles. I think this is interesting because usually during enlargments, the resulting extensions of sets not explicitly known and you may get a crazy number of additional points. The Noetherian property basically limits how crazy these additional points can get.
I'm sure I'm not the first to come across this characterization but I cannot find any resource on using nonstandard analysis to study noetherian rings, even in the context of algebraic geometry. Maybe I'm not looking in the right direction. If anyone can suggest some resources, or if you find this characterization interesting, can suggest how I should continue further in this nonstandard Noetherian ring direction, it would be much appreciated.