Question about light speed and relativity

[TechBear_Seattle]

I'm trying to work out some possible complications of relativistic sub-light travel for a story. Here is the scenario: A space ship is holding a position that is relatively stationary to you, 20 light minutes away. The beam from a comm laser -- a communications device using coherent light as a carrier wave -- will take 20 minutes to travel that distance.

Suppose the ship is moving directly away from you at 50% of the speed of light. If the laser is fired when the ship is 20 light minutes away, will it still take 20 minutes to reach the ship, correct? This despite the reasoning that for every one light minute the laser travels, the ship has gotten farther away by 30 light seconds. Likewise if the ship is moving directly towards you, correct?