Timeline for On serial Artinian rings
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2016 at 4:15 | answer | added | rschwieb | timeline score: 2 | |
| Sep 4, 2016 at 16:25 | comment | added | Luc Guyot | Quoting the wiki page en.wikipedia.org/wiki/Quasi-Frobenius_ring: Commutative Artinian serial rings are all Frobenius rings. You assumed $R$ Artinian in your question. | |
| Sep 4, 2016 at 13:51 | comment | added | Najmeh Dehghani | Thank you very much for your comment. In fact, I do not think a commutative generalized uniserial ring is Artinian Principal ideal. Beacue by Theorem 4.3 of the paper titled " On the decomosition of modules and generalized left uniserial rings", P. Griffith, we have the following theorem: A commutative ring R is Artinian Principal ideal ring if and only if R is generalized uniserial quasi-Frobenius. " So by this theorem, a generalized uniserial ring which is not quasi-Frobenius is not Atinian Principal ideal ring. Is my argument right? | |
| Sep 4, 2016 at 10:14 | comment | added | Luc Guyot | The wiki page en.wikipedia.org/wiki/Serial_module gives you the answer: By a result of Cohen and Kaplansky, a commutative ring RR has the property that its modules are direct sums of cyclic submodules if and only if RR is an Artinian principal ideal ring. By Nakayama, Artinian serial rings have this property on modules. So a commutative Artinian serial ring is a principal ideal ring (i.e., the answer is "no"). | |
| S Sep 4, 2016 at 9:58 | history | suggested | Luc Guyot | CC BY-SA 3.0 |
Corrected few typos: missing spaces, lower case for "serial" in the title
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| Sep 4, 2016 at 9:45 | review | Suggested edits | |||
| S Sep 4, 2016 at 9:58 | |||||
| Sep 4, 2016 at 6:19 | comment | added | Najmeh Dehghani | Sorry there was a mistake in my writing: " which is not principal ideal ring. | |
| Sep 4, 2016 at 6:16 | history | asked | Najmeh Dehghani | CC BY-SA 3.0 |