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We say that a commutative ring with unity is Noetherian if every nonempty family of ideals in the ring has a maximal element. It is known that, assuming the axiom of dependent choice, this is equivalent to the so-called ascending chain condition: every increasing denumerable ascending sequence of ideals is stationary.

Actually, the proof of this equivalence only uses the fact that the inclusion relation defines a partial order on the family of ideals in the ring, so it can be generalized to any poset (of course, assuming the axiom of dependent choice).

What always puzzled me is the fact that a "maximality" condition becomes just a "denumerable" condition. At least to me, "maximal" has to do with "potentially far larger than denumerable", and yet in this case the property reduces to a denumerable verification (and things "get worse" in the case of the Noetherian property, because another characterization is: any ideal is finitely generated, but in this case at least I can use the ring structure as scapegoat).

My question is: given an ordinal $\kappa>\omega$, what is equivalent to the statement "any ascending $\kappa$-sequence is stationary (in any poset)"? More importantly: Am I missing something in my reasoning above? Thanks in advance.

EDIT: I consider limit ordinals.

Matemáticos Chibchas
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In a partial order with the property that every increasing countable sequence of elements becomes stationary, then in fact every increasing ordinal sequence becomes stationary (indexed by a limit ordinal, otherwise it fails for trivial reasons). The proof is quite simple: If $(x_{\alpha})_{\alpha}$ is an increasing ordinal sequence, which is not stationary, then we can construct inductively (using DC) an increasing sequence of ordinal numbers $\alpha_n$ with $x_{\alpha_n} < x_{\alpha_{n+1}}$. But then $(x_{\alpha_n})_n$ is an increasing countable sequence which is not stationary, a contradiction.

Martin Brandenburg
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    The converse seems to be the interesting case. There can be partial orders where every increasing $\kappa$ sequence becomes stationary, but where there can be strictly increasing $\alpha$-sequences for some infinite $\alpha<\kappa$. (For example this is the case for $\kappa=\omega_2$ and $\alpha=\omega_1$ is the partial order itself). The question is then why (if?) such partial orders cannot arise as the family of ideals in a ring. – hmakholm left over Monica Mar 17 '13 at 20:13
  • @HenningMakholm I partially agree with you: I agree with the "interesting case" part, but any answer that settles the question is welcomed, not necessarily restricted to ring theory, or, at the opposite side, an answer that settles the question for a restricted class of order relations. – Matemáticos Chibchas Mar 17 '13 at 21:36
  • @Henning: They do arise, even for valuation rings. – Martin Brandenburg Mar 17 '13 at 22:21
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    @Martin: Then it seems that the question that puzzled the OP remains: How come the two definitions of "Noetherian" agree, if one is restricted to $\omega$ sequences and the other speaks absolutely about maximal elements without explicitly saying anything explicitly about the length of chains? – hmakholm left over Monica Mar 17 '13 at 22:36