Re: Ordinal numbers and rings


Arturo Magidin wrote:

Depends on the family! You should probably post the entire thing. I
suspect that this is much more a local issue than some sort of
underlying principle you seem to be groping for.



All right, allow me to post it. It's only a few sentence and I suppose
it's not difficult to understand if one knows how to deal with the
ordinals here.

This is a paper by Edgar Enochs on "Totally Integrally Closed Rings",
Proceedings of the American Mathematical Society, Vol. 19, No. 3 (Jun.,
1968), pp. 701-706

I will quote a statement on the first paragaraph of the proof on page
705:

----------------
suppose A is a subring of B, B is a limit ordinal number (-- maybe this
is a misprint, because I have this as an OCR scan of the original text,
so maybe he meant beta instead of B---- ), and (A_alpha) is a family of
subrings of B indexed by alpha<beta such that:

A_0 = A
A_(alpha+1) is a tight extension of A for all alpha<beta (--- tight
extension means that nonzero ideals of A_(alpha+1) if intersected with
A are nonzero ideals --- )
A_gamma = \bigcup A_alpha for alpha<gamma whenever gamma<beta is a
limit ordinal.

Then it is easy to check that \bigcup A_alpha for alpha<beta is a tight
extension of A.
-----------------------

The text before the one I just quoted isnt relevant, at most it is
assumed that B is a tight extension of A.. but I don't think being a
tight extension has anything to do with the construction that is being
made... I just want to understand the construction.. and the last
sentence which apparently assumed that \bigcup A_alpha is a ring... do
I need to get myself of a book on modern set theory to understand this?
I understand basic operation between ordinal numbers, how they are
added or even multiplied but thats as far as I know and when I look at
university scripts on set theory I don't see anything that I a don't
know regarding ordinal numbers that may help understand the text that I
just posted. Thanks in advance.

Sincerely,
Jose Capco

.

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