There are two ways to do a nuclear rocket. The "clean" way, closed loop. And the "dirty" way. Open loop. Closed loop is actually not all that different from us

There are two ways to do a nuclear rocket. The "clean" way, closed loop. And the "dirty" way. Open loop. Closed loop is actually not all that different from using a reactor as a generator. But instead of using your cooling system to power a turbine via a head exchanger, you use it to heat a reaction mass such as H2. Open loop is closer to what you describe. Instead of indirectly heating your reaction mass, you run it directly through the hot reactor. It's simpler and lighter but your exhaust is extremely radioactive. Your reaction mass essentially doubles as the cooling system. Another downside to this approach is you cant run the reactor without making thrust. So lights out if you are coasting I guess.