I'm having trouble with what seemed like a trivial problem at first, but now I'm rather stuck. If R is a ring with xy=-yx for any x,y from the ring, xyz+xyz=0 must be true for any x,y,z from the ring. I'm trying to show why that is.(adsbygoogle = window.adsbygoogle || []).push({});

Letting y=x yields x^2+x^2=0. Thus then breaking it up into two cases: x=0 (in which case xyz+xyz=0 trivially), and if not - then I get x^2z+x^2z=0. But I'm not sure if I can do this (as this doesn't necessarily show that xyz+xyz=0 but just that x^2z+x^2z=0) just because we can say y=x in some cases it doesn't seem like we can generalize that to all cases. Am I missing something simple or is there some other way of going about it? Any ideas would be much appreciated.

Also, I'm trying to find some information on the small Frobenius (basically the proof about why the map from x -> x^c is an automorphism where c is the char(F) for finite field F). I haven't been able to find anything very succinct or understandable on this.

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# Ring Question, Frobenius

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