Timeline for $\mathbb{Q}/\mathbb{Z}$ decomposition and possible generalization
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 30, 2022 at 11:57 | comment | added | Matthé van der Lee | This trivially holds if $R$ is semi-local (finitely many maximal ideals). And for Dedekind domains (assuming $\text{ann}_R(x) \ne 0$, of course). These are one-dimensional. | |
| May 30, 2022 at 10:37 | comment | added | Aleksei Kubanov | I'm curious about a condition that $\mathrm{ann}_R(x)$ is contained in only finitely many max. ideals. Are there any cases when this condition is satisfied automatically? What about one-dimesional domains in particular? | |
| May 30, 2022 at 10:30 | vote | accept | Aleksei Kubanov | ||
| May 30, 2022 at 10:28 | comment | added | Aleksei Kubanov | Thank you! I guess, in case $\mathfrak{p}$ is maximal, we could argue that the ring $R / \mathfrak{p}^\alpha$ is local with max. ideal $\mathfrak{p} / \mathfrak{p}^\alpha$, thus it satisfies the UMP of localization, 'cause its invertible elements are precisely the ones not in $\mathfrak{p}$. Is that right? So I believe it makes sense to pose my question in context of one-dimesional domains, yes? | |
| May 29, 2022 at 18:37 | history | edited | Matthé van der Lee | CC BY-SA 4.0 | added 48 characters in body |
| May 29, 2022 at 18:31 | history | answered | Matthé van der Lee | CC BY-SA 4.0 |