Question: Suppose I have two variables $X$ and $Y$ with $Y>0$. Can the random variables $Y$ and $\frac{X}{Y}$ ever be uncorrelated, i.e.,
$$\mathbb{E}(X)=\mathbb{E}(Y)\mathbb{E}\left(\frac{X}{Y}\right).$$ This seems counterintuitive since both are dependent on $Y$. But I just want to make sure there is not some weird case I am not thinking of. Actually, I just realized as I type that this is true if $X$ or $Y$ is a constant. How about in cases other than constant $X$ or $Y$ then? I guess this question simplifies to what are the conditions for 
$$\mathbb{E}\left(\frac{X}{Y}\right)=\frac{\mathbb{E}(X)}{\mathbb{E}(Y)}?$$

Answer: 
One simple case where your identity is true: Let $Y$ be some nonconstant and positive RV, and let $X:=cY$ for some nonzero $c$. Then $E(X)=cE(Y)=E(X/Y)E(Y)$.
How about this: if $Y=X$ (that is, they represent the same random variable). Then $X/Y=1$ and $Y$ and $1$ are uncorrelated.