Timeline for Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?
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14 events
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| Aug 9, 2022 at 19:54 | comment | added | Onur Oktay | Perhaps paraphrasing didn't lower the level of difficulty, but I believe an expert's eyes can catch what amateurs like myself may miss. I value your opinion and input as before. | |
| Aug 9, 2022 at 19:54 | comment | added | Onur Oktay | After pondering a while, I'm convinced that the problem is difficult in this generality. In a naive attempt to find the subsets of $\mathcal{S}$ that are closed under addition, let $E_m = E\cap maxspec(R)$ for $E\subseteq spec(R)$ and let $\mathcal{S}^* = \{E\in spec(R): k(E) = k(E_m)\}$ where $k(E)$ is the intersection of the prime ideals in $E$. $\mathcal{S}^*$ is closed under unions, so a slightly weaker version of Q1 above is: classify the maximal subsets of $\mathcal{S}^*$ that are closed under intersection. | |
| Aug 9, 2022 at 16:11 | comment | added | Keith Kearnes | @OnurOktay: About the questions in the body (For which classes of rings $\ldots$), I think that these questions are too hard if you want to consider arbitrary classes. If $\mathcal C$ is a class of commutative unital rings that (i) is closed under the formation of products and subrings and (ii) contains a field that has a valuation subring that is not a subfield, then I think you can mimic the above construction within $\mathcal C$. This rules out a lot of classes. | |
| Aug 9, 2022 at 16:06 | vote | accept | Onur Oktay | ||
| Aug 9, 2022 at 16:06 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 15:53 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 15:39 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 15:29 | comment | added | Onur Oktay | I'd like to note for the readers like myself that the construction has immediate (made obvious by the stages) generalization if one starts with an integral domain and a prime ideal different from $D=\mathbb{Z}$ and $P=2\mathbb{Z}$ respectively. $L$ is the localization $D_P$. $S=c_{00}(\mathbb{N}\to D_{{0}})\oplus\mathbf{1}L$, where $\mathbf{1}=(1,1,\dots)$. $c_{00}$ could be substituted with another ring of sequences without $\mathbf{1}$ in it, e.g., $\ell^p(\mathbb{N}\to\mathbb{Q})$ for $D=\mathbb{Z}$. | |
| Aug 9, 2022 at 15:29 | comment | added | Onur Oktay | Professor @KeithKearnes, thank you for your helpful approach and extensive reply. Your construction is work of art in my humble opinion. Thanks also for all the details and explanation - words are like salad, help to digest the main course, clearly the more the merrier. $R$ is indeed a counterexample to the question in the title. May I ask what's your insight for the two questions in the body? Perhaps I should post these two in a separate question. | |
| Aug 9, 2022 at 7:46 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 7:40 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 7:14 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 5:16 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Aug 9, 2022 at 4:52 | history | answered | Keith Kearnes | CC BY-SA 4.0 |