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Jean-Baptiste Vienney (Feb 12 2024 at 19:02):I'm interested in Prüfer rings i.e. commutative rings such that every nonzero finitely-generated fractional ideal is invertible.
It is claimed here that Prüfer rings are coherent and that the definition of coherent ring implies that the intersection of any two finitely-generated ideals of a coherent ring is finitely generated (here, they even claim that the notion of coherent ring is equivalent to this condition).
Finally, whatever is or how should be defined a coherent ring, I'm only looking for a proof of this precisely defined proposition: "the intersection of two finitely-generated ideals of a Prüfer ring is finitely-generated".
Does anyone know about this?
(I'm interested in this because it seems that the (opposite) of the lattice of nonzero finitely-generated fractional ideals of any Prüfer semiring is a perfect model of some sequent calculus for arithmetic that I'm working on. This fact looks very important in this topic.)
Todd Trimble (Feb 12 2024 at 22:52):Here's a sketch: invertible modules are projective, and finitely generated projective modules over a commutative ring are finitely related (there's a surjection whose kernel is finitely generated; that's clear because you can split any exact sequence , and then surjects onto ). Thus, in a Prüfer domain, finitely generated ideals are finitely related.
Then refer to Theorem 2.2 here, page 6 of 17, particularly the implication (a) => (c) where this gives that if every finitely generated ideal is finitely related, then the intersection of any two finitely generated ideals is also finitely generated.
Jean-Baptiste Vienney (Feb 12 2024 at 23:03):Thank you very much @Todd Trimble !
Jean-Baptiste Vienney (Feb 12 2024 at 23:10):I'm still not sure if what I call a Prüfer ring must be called a Prüfer ring. I was just reading two minutes ago that there are several nonequivalent extensions of the notion of Prüfer domain to rings with zero divisors. But you made a lot of work for me by giving a sketch of why Prüfer domains verify the intersection property. That's some good progress :)
Todd Trimble (Feb 12 2024 at 23:16):Well, I'm not at all versed in all this. But that theorem I pointed to seems to contain answers to the other questions you ask, to the effect that a ring is coherent iff the intersection of any two finitely generated ideals is again finitely generated.
Todd Trimble (Feb 12 2024 at 23:19):Interestingly @Morgan Rogers (he/him) was also asking about related things some years back, here. It was through a comment under his question that I was alerted to the paper by Stephen Chase that I linked to.
Jean-Baptiste Vienney (Feb 12 2024 at 23:47):I saw also that Morgan asked similar questions on Math StackExchange ahah. People are interested by the good stuff for different reasons ahah.