On diagonalizability of regular matrices over rings

Some results on the problem of the diagonalizability of an arbitrary von Neumann regular matrix over an associative ring with unit are proved. There are constructed two examples of rings which refute the next conjecture of J. Van-Geel and D. Huylebrouck: if $R$ is an ID-ring (i.e. all idempotent matrices over $R$ are diagonalizable) then every von Neumann regular matrix over $R$ is diagonalizable.