Functional equation on a ring

Question

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and \begin{equation} f(x+y, x-y)=0 \end{equation} for all $x, y\in R$. If $R$ is a field, or $f(2, 0)\ne 0$, then it can be shown that $$ f(x, y)=M_1(x+y)M_2(x-y) $$ for all $x, y\in R$, where $M_j(xy)=M_j(x)M_j(y)$ for all $x, y\in R$ and $j=1, 2$.