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Let $A$ be a commutative ring. Let $f$ be any non-zero element of $A$. Suppose that $A/fA$ has a composition series as an $A$-module. Then we say $A$ is a weakly Artinian ring (this may not be a standard terminology).

Can we prove the following theorem without Axiom of Choice?

Theorem Let $A$ be a weakly Artinian integrally closed domain. Then the following assertions hold.

(1) Every ideal of $A$ is finitely generated.

(2) Every non-zero prime ideal is maximal.

(3) Every non-zero ideal of $A$ is invertible.

(4) Every non-zero ideal of $A$ has a unique factorization as a product of prime ideals.

EDIT May I ask the reason for the downvotes? Is this the reason?

EDIT What's wrong with trying to prove it without using AC? A proof without AC is constructive. When you are looking for a computer algorithm for solving a mathematical problem, this type of a proof may provide a hint. At least, you can be sure that there is a constructive proof.

EDIT why-worry-about-the-axiom-of-choice.

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    Are you going to ask a single question for each and every combination of algebraic properties, whether or not it holds without the axiom of choice?2012-07-14
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    You can prove without AC that every ideal of $A$ is contained in a maximal ideal (take a chain - it terminates, cause all chains do because of the composition series ---> no reason to appeal to Zorn's lemma) and so I think the answer to all four questions is affirmative.2012-07-14
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    @MakotoKato I see that many of your posts have drawn a lot of flak from users on math.stackexchange. I think that the people at MO would be more competent to handle your questions, why don't you post them there on MO? I can help you by posting them on your behalf on MO if you would like to. Regards,2012-07-14
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    @BenjaLim: I'm not sure that the good folks on MO will appreciate this behavior of posting a convergent sequence of partial answers. If anything I would expect mathematicians to appreciate the ability to hold back the answer *until* it was about done. I also don't know if people would appreciate a rain of questions verifying a majority of commutative algebra is constructive for "tame" objects.2012-07-14
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    @AsafKragila "Are you going to ask a single question for each and every combination of algebraic properties, whether or not it holds without the axiom of choice?" No. I ask it mostly when the answer is likely to be affirmative and mostly when I think it's interesting and it has important applications.2012-07-14
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    @Asaf, do you have some kind of objection in principle to asking about the various uses of AC in algebra? To my way of thinking, this is a perfectly reasonable topic of study, particularly in those cases where it seems likely that AC is required.2012-07-15
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    @Makoto Please do not bump questions or answers for the purpose of asking about downvotes. Such questions are only rarely fruitful. If you insist on posing such questions then please do so in comments, so that the thread is not bumped. Better, as always, discuss such *meta* issues on the meta site - where they belong.2012-07-16
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    @BillDubuque I'll try to put them as comments from now on. However, I can't promise I'll always do so. I may put them in my qestions or answers in rare cases. Is that okay?2012-07-17
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    @YACP, please do not bring back questions from the dead for the sole purpose of italicizing a phrase. I also prefer terms being defined to be emphasized in some way, but it is not *that* important...2012-12-30
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    @YACP, well, it may be important for you, but there is a scarce resource which is the list of Top Questions, which is *shared* between all users, those that have absolutely no interest in this sort of edits and you. It is not polite to proceed under the assumption that your interest in italics trumps other's people questions time share on the main list.2012-12-30
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    The point is, you are contributing mostly *noise* to the site by doing this. Surely you understand this... If you intend to keep this up, please be nice and start a meta thread to see what the community thinks about your quest for typographical correctness.2012-12-30
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    Well, I am trying to moderate your bumping old questions for the sole purpose of correctly typographical errors!(actually, to be precise: what *you* see as errors, as there is no International Treaty one should respect by italicizing defined terms, really... It may well be the case that MK purposefully did not italicize those terms because *he* has a different idea of what «perfect» is...) I suggest that you strive to perfect what you write, and leave minor would-be-issues like these on other people's post alone.2012-12-30
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    **Please** make any further comment on this topic in a meta thread: this is 200% off-topic here.2012-12-30

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