Talk:Supercommutative algebra

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[edit] Definition with 2-torsion

I notice that R.e.b. made a correction to this article sometime ago regarding the square of odd elements. With the present definition we have

2x² = 0

for all odd x. If there is no 2-torsion this implies that

x² = 0.

However, it seems to be an crucial fact when dealing with commutative superalgebras that the odd elements square to zero. In all of the references I can find, authors either (implicitly or explicitly) assume that 2 is invertible or that there is no 2-torsion, or they explicitly require that x² = 0 for all odd x. I may edit the article to this effect, but I'm interested in what others have to say. -- Fropuff (talk) 22:03, 7 February 2008 (UTC)

Actually, the more I think about, modifying the definition of supercommutative to insist that x² = 0 for all odd x seems like the wrong thing to do. It would no longer be true that a superalgebra is commutative iff it is equal to its opposite, or to its supercenter, or iff the supercommutator vanished identically. I guess the correct thing to do is simply assume that 2 is invertible whenever necessary (or at least assume there is no 2-torsion). -- Fropuff (talk) 08:26, 8 February 2008 (UTC)