Timeline for Does Smith normal form imply PID?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| S May 3, 2017 at 23:01 | history | suggested | user26857 | CC BY-SA 3.0 |
fixed a typo
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| May 3, 2017 at 22:40 | review | Suggested edits | |||
| S May 3, 2017 at 23:01 | |||||
| Jul 10, 2010 at 14:55 | comment | added | Manny Reyes | A commutative noetherian ring whose maximal ideals are principal is indeed a principal ideal ring (even if it is not domain). See Theorem 12.3 of Kaplansky's article "Elementary divisors and modules," Trans. Amer. Math. Soc. 66 (1949), 464-491. | |
| Jul 10, 2010 at 7:19 | comment | added | Robin Chapman | Thanks a-fortiori: each localization at a maximal ideal is a local ring of height at most one, so $R$ has Krull dimension $\le 1$. | |
| Jul 10, 2010 at 7:13 | comment | added | user2035 | For R a domain, it implies that R is one-dimensional regular, hence Dedekind, so every nonzero ideal is a product of maximal ideals, therefore principal itself. | |
| Jul 10, 2010 at 6:47 | history | answered | Robin Chapman | CC BY-SA 2.5 |