Let $\odot$ denote the standard multiplication operation: $a\odot b=ab$. Which of the following distributive laws hold for all numbers $x$, $y$, and $z$?

I. $x\odot (y+z)=(x\odot y)+(x\odot z)$
II. $x+(y\odot z)=(x+y)\odot (x+z)$
III. $x\odot (y\odot z)=(x\odot y)\odot (x\odot z)$