The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)

The statement in the title is not generally true, unless $C$ and $R$ have full rank. Then the $m$ by $r$ matrix $C$ is assumed to have $r$ independent columns (rank $r$). The $r$ by $n$ matrix $R$ is assumed to have $r$ independent rows (rank $r$). In this case the pseudoinverse $C^+$ is the left inverse of $C$, and the pseudoinverse $R^+$ is the right inverse of $R$