Theories

On the Speed of Light

So I was doing some research for my English paper, and I was reading a book about the theory of relativity. That got me thinking about Einstein’s “Riding on a ray of light” idea, and what if the ray of light was shining, being mounted on something moving? Would the speed of the light change? If the object was moving forward, and the light was shining forward, wouldn’t the ray of light travel faster than the speed of light? And if the object was moving opposite of the shining direction, would the ray be moving slower? Now suppose that the object was traveling at near light speed, within a couple 1000km/s of it. Would the light travel by you, if the object was moving away from you, while the light was shining back? And shined the opposite way, the light would relatively be moving very slow. This may have been some other idea, that was thus solved in the theory of relativity or something of the sort, but nothing I can recall can explain it.