1. You don't need to conjugate if you want $R$ to be diagonalizable (as opposed to diagonal).

2. I assume you want $R$ and $M$ to have coefficients in $K$, otherwise just work in the algebraic closure.

3. The statement is then true if $K$ is perfect and possibly false otherwise, as you can see by taking $A=[[0, 1], [t, 0]]$ in $K=F_2(t)$.

1. You don't need to conjugate if you want $R$ to be diagonalizable (as opposed to diagonal).

2. I assume you want $R$ and $M$ to have coefficients in $K$, otherwise just work in the algebraic closure.

3. The statement is then true if $K$ is perfect and possibly false otherwise, as you can see by taking $A=[[0, 1], [t, 0]]$ in $K=F_2(t)$.

EDIT : see http://en.wikipedia.org/wiki/Jordan%E2%80%93Chevalley_decomposition