Understanding the definition of Schnorrs identification protocol I have some trouble in understanding the definition of Schnorr's Identification protocol. The is the definition that I received:

Prover and verifier agree on a cyclic group $G$ of prime order $q$ generated by $g$.


The prover sends $r=g^k$ for a random $k$
The verifier sends $e \in_R Z_q$
The prover sends $s=k+xe$
The verifier now verifies if $r=g^sy^{-e}$


Now I don't understand why $r$ and $e$ have different formulas. As far as I understand, $r$ is an element of the cyclic group $G$, and $e$ also. What is the difference between: 

$\in_RZ_q$

and 

$g^k$ for a random $k$

mean?
The difference is the amount of information you have about the structure of the numbers.

Let $e\in_R\mathbb Z_q$. Then you know that the $e$ you have, is a random number from that set. You know no more and no less.

Let $r=g^k$ for some $k\in_R\mathbb Z_q$. Then you know that the $r$ is a random number from the group and you know its discrete logarithm to a pre-defined element. The fact that you randomly picked $k$ doesn't matter, you still know it and can use it in step 3. Even better you know that it is highly unlikely that somebody else will be able to infer $k$ from $r$ (because that is the discrete logarithm problem).

So you see: You know a lot more about $r$ than about $e$ because it was generated in this slightly different way.