Timeline for Artin Jacobson-semisimple rings are semisimple. Constructively, too?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2015 at 15:42 | answer | added | Pace Nielsen | timeline score: 6 | |
| Jan 26, 2015 at 13:16 | history | edited | darij grinberg | CC BY-SA 3.0 |
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| Jan 26, 2015 at 13:09 | history | edited | darij grinberg | CC BY-SA 3.0 |
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| Jan 26, 2015 at 4:17 | answer | added | Pace Nielsen | timeline score: 0 | |
| Dec 16, 2011 at 21:22 | comment | added | darij grinberg | Anyway, any proof you give that is "more constructive" than that in Lam (i. e., no use of the AC, or more algorithmic content) is bound to get upvotes (at least from me). | |
| Dec 16, 2011 at 21:21 | comment | added | darij grinberg | ... In a nutshell, "constructive proof" means a proof that gives an algorithm (which requires only finite time) which takes any $r\in R$ and gives back an $x\in R$ satisfying $rxr=r$; it can (and will have to) use a function (which is considered to be already implemented) that takes an increasing sequence $I_0\supseteq I_1\supseteq I_2\supseteq ...$ of ideals (i. e., an algorithm to construct $I_n$ for every $n$, a proof that it is indeed an ideal, and a proof that $I_n\supseteq I_{n+1}$) and gives you an $n$ such that $I_n=I_{n+1}$ (but no other oracles). | |
| Dec 16, 2011 at 21:18 | comment | added | darij grinberg | @rschwieb: I define the notion of Artinian rings by the definitions given above. If you can show it to be constructively equivalent to some other one, then it's great and it would be a big surprise for me. As for von Neumann regularity, I don't know of any other equivalent definitions, but I tend to believe that if they are some, they are most likely constructively equivalent to mine. ... | |
| Dec 16, 2011 at 20:39 | comment | added | rschwieb | Apologies in advance, because I am not familiar with what is not allowed in constructive proofs. Are we to use only the definitions above, without the usual alternative characterizations of VNR and Artinian semisimple rings? | |
| Mar 9, 2011 at 0:36 | comment | added | darij grinberg | Classically it is equivalent to standard Artinian, and constructively standard Artinian makes pretty much no sense. And as I am apparently the first constructivist to care about noncommutative rings, I decided to take the simple way and just call it Artinian. I am more or less aping Richman in math.fau.edu/richman/Docs/rep.ps . | |
| Mar 9, 2011 at 0:19 | comment | added | Mariano Suárez-Álvarez | You could pick another name, as reusing the usual one for something different and (hopefully...) related, can only cause pain! Why not "constructively left Artinian" or something? | |
| Mar 8, 2011 at 23:15 | comment | added | darij grinberg | Yes, I want to work with my definition of Artinian - unless somebody gives me a better one (which I hope). The usual one is utterly useless in constructive mathematics. | |
| Mar 8, 2011 at 23:08 | comment | added | Martin Brandenburg | So you really want to work with the above definition of artinian, which is not the usual one at first sight? By the way I've deleted the tag "von Neumann algebras" since they are special $C^*$-algebras completely unrelated to von Neumann regular rings. | |
| Mar 8, 2011 at 23:07 | history | edited | Martin Brandenburg |
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| Mar 8, 2011 at 21:54 | comment | added | darij grinberg | By "classical Artinianity" I mean what MTS has written. Constructively it is a FAR too strong condition (not even the field with two elements is constructively classically Artinian, because this would solve the halting problem). | |
| Mar 8, 2011 at 21:46 | comment | added | Manny Reyes | I'm just curious, by "classical artinianity" do you mean the condition that every set of left ideals has a minimal element? | |
| Mar 8, 2011 at 21:26 | comment | added | MTS | Not to be nitpicky, but (here begins nitpickiness) in your definition of left and right Artinian, you need $I_n = I_{n+1}$ for all $n \ge M$ for some $M$. | |
| Mar 8, 2011 at 19:31 | history | asked | darij grinberg | CC BY-SA 2.5 |