## 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.