Can the projective line be provided with a ring structure? A definition of multiplication on the projective $1$-points $(a:b)$ of $P_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the multiplication of complex numbers.
Is it known whether there a way to define an addition between projective $1$-points ? 
 A: The topological space $\mathbb{P}^1_{\mathbb{R}}$ is a circle, so it has an abelian group structure just defined by adding angles but this is not algebraic, not compatible with the multiplication you describe, and it can't be extended to $\mathbb{P}^1_{\mathbb{C}}$ (which is a sphere and thus has no topological group structure), nor does it restrict to $\mathbb{P}^1_{\mathbb{Q}}$. EDIT: As Noam Elkies points out,  the circle group structure can be made algebraic on $\mathbb{P}^1_{\mathbb{R}}$ by defining it by $\tan(x+y)=\tan(x)+\tan(y)$, instead of by adding angles.  This still has all the other defects listed above.  
More generally, it's impossible to do this algebraically: it's an extremely well known fact that a projective curve with an algebraic group structure (as addition would be) must be an elliptic curve.  
A: Here is a less algebraic and more topological answer: it's known that any compact (Hausdorff) topological ring must be totally disconnected. In particular, there's no hope for either $\mathbb{RP}^1$ or $\mathbb{CP}^1$ to have the structure of a topological ring. (As Ben Webster mentions, $\mathbb{CP}^1$ also can't even be a topological group.) 
