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2014-10-16

Blindsight spot.



I just realised that in any unital ring commutativity of addition follows from distributivity:

\(a + b\)
\(=\)\(-a + a + a + b + b - b\)
\(=\)\(-a + a\cdot(1 + 1) + b\cdot(1 + 1) - b\)
\(=\)\(-a + (a + b)\cdot(1 + 1) - b\)
\(=\)\(-a + (a + b)\cdot 1 + (a + b)\cdot 1 - b\)
\(=\)\(-a + a + b + a + b - b\)
\(=\)\(b + a\)

The same holds for unital modules, algebras, vector spaces, &c. Note that multiplication doesn't even need to be associative. It's amazing how such things can pass unnoticed.

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