Grayscale to colour

I'm not quite sure how to phrase this, which is also why I haven't been able to search for an answer.

What I'd like to do is, have a grayscale image and then convert the lightness values of that image into lightness values of a given hue, if that makes any sense.

Such that if I choose some red hue, then the areas that are 100% black will be 100% red, and an area that is 50% black will be red with 50% white in it.

Basically like if I had fed the grayscale image to a printer with only red ink, and it then used the lightness values of that image to decide the ink density of the different areas.

Is this possible in photoshop?

Thanks!