Suppose $R$ is noncommutative ring with unit and has the properties necessary for a right and left skew field of fractions to exist (i.e. $R$ has no zero divisors and satisfies the left and right Ore condition). Let $x,y\in R$ be nonzero elements in $R$ so that $x,y$ are prime and $xy=yx$ in $R$.
I am interested in showing that the solutions to the equation $$xp+yq=0 $$ are exactly $p=yt, q=-xt$ for all $t\in R$.
Here is my reasoning:
$p=q=0$ is a solution. Now consider all other solutions.
If we assume one of $p$ or $q$ is nonzero, then since there are no zero divisors in $R$, we know both $p$ and $q$ are nonzero. In the skew field of fractions we can rewrite the equation as $$p=-x^{-1}yq $$ but since $x$ and $y$ commute, we have $$p=-yx^{-1}q. $$ Since the left side resides in $R$ if seems like $q=xt$ for some $t\in R$, since $y$ can not contain a factor of $x$ by the primeness assumption. By a similar argument, we can conclude that $p=ys$ for some $s\in R$. Hence the equation becomes $$xys+yxt=0 .$$ Multiplying on the left by $x^{-1}y^{-1}$ yields $s=-t$ Proving the result.
I feel I may be making a mistake. Some of the steps may not work in such a general setting? Thanks.