Doors and Windows - doors

I have a hollow core door going from my kitchen to my basement. The door opens into the kitchen and was terribly inconvenient, so I removed it. That is ok for now, but I'd like a 1/2 door there. Is there any way to cut the door in half and then put something in between the core of the door so that it can still be used, or do I have to buy a specific 1/2 door?

Yes, you should be able to cut the door in half with a fine-tooth paneling saw blade, and add a 2X piece across the middle and move the other hinge down to the top of it. Just saw it off above the existing doorknob.
I had a friend that just built a half-door with thin plywood and 2X framing. Same thing. Works fine.
Good Luck!

Thanks for your help, but I'm kind of construction illiterate, what do you mean by 2x? Is that what I use to fill the hollow door with?

I just meant a 2"X4" (actual measurements: 1.5" thick X 3.5" wide) board split down to fill in the open top of the door half between the side frames, after you saw it in half. Most hollow core doors are about 1-1/8" between the panels, so you have to rip it down from 1.5" to 1-1/8" on a table saw.
You can dress it up with a cap board over the top of that, if you want, and quarter-round molding under the cap.
Paint or stain to suit.
Just measure the thickness of your door and you'll see what I mean.
I've never tried it, but you may be able to take apart the upper part of the door that you saw off, and use a part of the frame that's already the proper thickness.
If it were me, I think that I'd just store that door in the basement, and build a half-door with a 2X4 or 2X6 cross-braced frame and 1/4" plywood or 1/8" paneling. Just add two hinges and a doorknob lockset. Then if you ever decide to replace it, you just re-hang the original door. Your call.
Good Luck!