1

I am trying to clarify in my head the different meanings of "ideals" in mathematics. We have ideals in Number Theory, as in Dedekind (derived from 'Ideal Complex Numbers' in Kummer), in Abstract Algebra (Ring Theory), as in Dedekind and, mostly, in Noether's development, and we also have, it seems, a notion of ideals in Order Theory, in which they are a special subset of a partially ordered set (poset). I have read the meaning in order theory developed from the second meaning (that in abstract algebra and ring theory).

I would like to know:

1) How it precisely developed 2) Who is responsible for such development (Author, references, original definiton and so on). Wikipedia credits Marshall H. Stone with that. Is that right? Any references and quotations from the original work?

Thanks in advance.

Javier Arias
  • 2,033

1 Answers1

0

One of key concepts in Order Theory is duality. Given that this is so, and filters are duals of ideals, it may be worth your while to trace back the developments with regard to filters and ultrafilters, as those have often been the primary motivation of research. In many cases, results concerning ideals may have just emerged by default, as offshoots of effort placed in proving results concerning their duals, i.e. filters.

Mariusz Popieluch
  • 682