I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-0031470-3.pdf) where he gives a ``local" characterization for a noetherian ring to be a principal ideal ring. More precisely, the result he proves (in Theorem 12.3) is that any noetherian commutative ring $R$ with all maximal ideals principal must be a PIR and furthermore must be a finite direct sum of integral domains and Artinian valuation rings. The proof divides into two cases:

  1. In the first case, he assumes that all the zero divisors of $R$ are contained in the Jacobson radical in which case he shows that the sum of two principal ideals $aR$ and $bR$ is principal (since $R$ is Noetherian, it is sufficient to show this). To that end, he assumes that if $aR + bR$ is contained in some maximal ideal $cR$ (otherwise the sum is the whole ring and we're done) then one obtains principal ideals $a_1R$ and $b_1R$ properly containing $aR$ and $bR$ respectively. Then he considers the ideal $a_1R+b_1R$ and so on, and this must be a finite process since $R$ is Noetherian.

  2. In the general case, he considers the irredundant primary decomposition of the zero ideal and uses his Lemma 12.2 to pass to the primary components (or rather the complements of the primary components, which have all their zero divisors in their respective radicals) by writing $R$ as a sum of any two of them. This reduces to 1 and simultaneously yields the structure theorem sought.

Now I was curious as to whether (in the general case) it is possible to give an independent proof of the fact that $R$ is a PIR, perhaps without saying anything about its structure. I couldn't find a simple proof of this partial result on MO or MSE, but it seems to me that one should be able to simplify the arguments and give a shorter proof of just the first part of Kaplansky's result, although I haven't personally succeeded in getting anything non-trivial to that end. Has this been done in prior literature? I would be grateful for any references, proofs, hints, suggestions or ideas.



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