Q:

A Rng of rotations?

It's very standard to view rotations about the origin in $\mathbb{R}^2$ as a group $\mathbb{SO}(2)$, with the zero rotation as an identity and composition of rotations as addition. This can also be viewed as the interval $[0,2\pi)$ with $0$ as the additive identity and addition defined mod $2\pi$.
We can lift this to a Rng structure by defining multiplication $\hat\times:[0,2\pi)\times[0,2\pi)\rightarrow[0,2\pi)$ as $$a\hat\times b=\frac{ab}{2\pi},$$ so each rotation is treated as a fraction of a complete rotation of $2\pi$. Since $a,b<2\pi$ we have that $\frac{ab}{2\pi}\leq\min\{a,b\},$ with equality holding iff $a=0$ or $b=0$. This is a rng and not a ring because the multiplicative identity we would like, $2\pi$, cannot be allowed in the set on consequence of non-uniqueness for the additive identity.
Is this a well known construction, or obviously trivial as a geometric structure for some reason I'm missing? 

A:

The multiplication does not give a Rng structure because it does not distribute over addition:
$$
\pi\hat{\times}\big(\pi+\pi)=\pi \hat{\times} 0=0\neq \pi = \frac{\pi}{2}+\frac{\pi}{2}=\pi\hat{\times}\pi +\pi\hat{\times}\pi.
$$

