How far could one fire a charged particle in space? I've wondered how far would be able to send a concentrated beam of electrons in space. The reason is would we be able to launch a magnetic ring with an electron absorbing material on one side and just fire electrons through the ring to create a push? 
 A: Even in the "empty" space there are fields. In particular magnetic field is going to bend a trajectory of a charged particle. The Earth's magnetic field is strong enough to trap electrons of energies up to 10 MeV and protons with energies up to 100 MeV. The charged particles from solar wind and cosmic rays form Van Allen radiation belt but this magnetic field would also prevent electrons 'fired' at low Earth orbits from escaping.
Outside the Earth magnetosphere the magnetic field is generated by the currents around the sun, and though the fields are smaller (away from the Sun and other magnetic planets) it still will bend trajectories of the charged particles. So a precision 'shooting' with charged particles would be very difficult.
Nevertheless, there is a proposed system of space propulsion, the electric sail, which is somewhat similar to your suggestion. There, positively charged wire is used to deflect protons from the solar wind to produce the thrust.
A: Firing particles with a net charge to create a push doesn't work in the long run.  If you continuously fire electrons, then your ship will become positively charged.  This charge gets larger and larger as more electrons get emitted.  Eventually all the electrons you fire will "fall" back onto the ship so that you get no net push.
To get a sustained push from firing particles, make sure to fire equal charges so that no net field is being created.  For example, create a hydrogen plasma and then separately accellerate the electrons and the protons.  You can accellerate them in the same direction so that you get a net push without a net charge buildup.
