Timeline for On a theorem of Jacobson
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21 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Jul 24, 2010 at 19:08 | history | edited | Victor Protsak |
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| Jul 15, 2010 at 21:13 | answer | added | T.. | timeline score: 6 | |
| Jul 15, 2010 at 20:38 | comment | added | Kevin Buzzard | It still might be true though! I certainly don't know a counterexample (as you probably guessed---because if I did I would have played it instantly!). I am not optimistic about finding a "nice" maximal set though. I think the first thing to do is to read the proof and to see what's really going on, and to go from there. | |
| Jul 15, 2010 at 20:09 | comment | added | Pierre-Yves Gaillard | Dear Kevin: I fully agree with you! My previous two comments were completely silly! Sorry about that ... | |
| Jul 15, 2010 at 19:59 | comment | added | Kevin Buzzard | @Pierre: that, for me, is not a proof. The theorem, as I understand it, is that if for all $r$ there's $n$ with $r^n=r$ then $R$ is commutative. But this $r^n=r$ thing is a huge extra constraint on the ring, that most commutative rings do not come close to satisfying. It shows, for example, that the ring has finite characteristic (set $r=2$). If you remove this trick (because $r=2$ is no longer allowed) can you still prove that $R$ is commutative? You know more about the proof than I do, I'm sure, but I don't think your comment is enough---it's certainly not enough for me, at least! | |
| Jul 15, 2010 at 19:38 | comment | added | Pierre-Yves Gaillard | Dear Kevin, I think my set works because the prime ring is always central. | |
| Jul 15, 2010 at 19:24 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
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| Jul 15, 2010 at 19:23 | comment | added | Kevin Buzzard | @Pierre: can you prove that your set works before we start worrying about whether it's maximal? I only added $X-9$, I didn't add all $X-n$ simultaneously. But my gut instinct is that, if your set is OK, it won't be maximal because there are still plenty of other stupid tricks you can try (square of a linear factor etc). Note however that if someone comes up with an enlargement and then someone else says "OK then is this enlarged set maximal" we could be here all year! | |
| Jul 15, 2010 at 18:53 | comment | added | Pierre-Yves Gaillard | Let's add to $S\cup-S$ the degree one monic polynomials, the divisors of the $X^n-X$, and the negatives of the above. Is this enlarged set maximal? | |
| Jul 15, 2010 at 18:17 | comment | added | Kevin Buzzard | [clarification: "divisor" means "divisor in $\mathbf{Z}[X]$"] | |
| Jul 15, 2010 at 18:17 | comment | added | Kevin Buzzard | $S$ isn't remotely maximal, as far as I can see. For example any divisor of $X^n-X$ for any $n$ can be added to it, as if $P(X)=0$ for $P$ some divisor of $X^n-X$ then $X^n-X=0$. Moreover, if you have a ring in which every element other than the number 9 satisfies $X^n=X$ for some $n$, then 9 will also satisfy this, because $3^n=3$ implies $9^n=9$. So you can also add $X-9$ to $S$. And so on and so on... | |
| Jul 15, 2010 at 18:03 | comment | added | O.R. | and x-1 can be also added to S | |
| Jul 15, 2010 at 17:08 | comment | added | Daniel Litt | Ah I see, you're right of course. | |
| Jul 15, 2010 at 16:57 | comment | added | Pierre-Yves Gaillard | Dear Mariano: Thanks for mentioning this question. I think one should ask if $S\cup(-S)$ [and not $S$] is maximal for the property in question. | |
| Jul 15, 2010 at 16:56 | comment | added | O.R. | @Litt, I guess you are taking minimal in cardinality. But minimal in what sense, such that it implies commutative? x^2-x=0 is enough. Notice that putting more elements in the set doesn't necessarily make things better to get the commutativity because then some a's can satisfy equations from S and some others satisfy the new equations. (or all satisfy the new equations). | |
| Jul 15, 2010 at 16:51 | comment | added | O.R. | If S works then a subset of S also work. Maximal is the right question. | |
| Jul 15, 2010 at 16:50 | comment | added | Daniel Litt | Any set containing S seems to work. Perhaps you should edit the question so it asks the classification question in you comment. (E.g. classify minimal sets S with this property). Note that S is not minimal either; consider $X^{n!}-X$ for example. | |
| Jul 15, 2010 at 16:38 | comment | added | Mariano Suárez-Álvarez | Of course, one can very easily become greedy and start asking questions like «can one classify sets with that property?» and so on. This looked like a good start :) | |
| Jul 15, 2010 at 16:37 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
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| Jul 15, 2010 at 16:31 | history | asked | Mariano Suárez-Álvarez | CC BY-SA 2.5 |