Can I run a sink drain between the joists in my basement?

I realize codes vary from place to place, so I’m just looking for a general answer.

Here’s my situation: Had a clog in my kitchen sink drainpipe. After using a clean-out to access a clog with a snake, the fitting cracked when I went to replace the cap. Went to my local neighborhood hardware store (the folks there are usually pretty knowledgeable) to get supplies to replace the clean-out. I cut the line (2” id ABS) above and below the existing clean-out; I then attached the new clean-out to a length of new pipe, and attached a coupling to the upper end of the new length. My intention was to attach the new clean-out directly to the existing pipe coming up from the concrete floor, and cut the existing line to fit the top of the new fitting. When I placed the fitting on the lower existing pipe, the metal elbow (in the concrete floor) gave way and broke! When I finally stopped kicking the wall in frustration over having to rent a jack-hammer and replace the in-floor line, I started wondering if I could just run the drain line through the ceiling, across the basement, and tie it into the drain line from my bathroom sinks. Is this something feasible, provided I have the right amount of slope? Both drain systems are currently vented; would this necessitate any more venting?