Let $R$ be a non-commutative ring and let $Rr$ be a maximal left ideal of $R$. Is it true or false that for every invertible $s\in R$ the ideal $Rs^{-1}rs$ will be a maximal left ideal?
My experiments with the ring $R = M_n(P[x])$ of matrices over polynomials over field show that this statement is true.
The map $r\mapsto s^{-1}rs$ is a ring automorphism of $R$. So it takes left ideals to left ideals and maximal left ideals to maximal left ideals. So if $Rr$ is a maximal left ideal, so is $s^{-1}Rrs$, but that is the same as $Rs^{-1}rs$.