Hidden electrical wiring questions can be difficult to explain. Being we not able to visually inspect your project providing good pictures and documentation along with some troubleshooting skills is important. Must provide as many details as possible when conversing. Scheduling a service call with a Licensed Electrician may be the end result.
investigate: Is the light fixture controlled by wall plate switch or by a push button switch on the fixture?
After some investigation: Possible neutral back feeding or incorrectly wired and shared on unknown 2nd-phase? This is one of the most dangerous situations in residential but could be totally different situation on commercial job.
This is a very serious situation! Get it fixed.
Existing ceiling surface-mounted pull chain broken so lite not switching on/off, so you want to use screw-in socket adapter with pull chain?
If you have a questions try not get your information from searches on the internet. If any of the above sounds familiar you need a licensed master electrician. These types of questions are questions people ask when they are leaning towards a diy repair. This is a very slippery slope and you may fall off. These questions are only a few of many asked before deciding to call a licensed electrician. In my experiences some of these questions are signs of a potentially extensive electrical problem. Unusual electrical wiring problems are usually worse than they look. If you are questioning wither to call a licensed electrician and any of these questions sound familiar please do not make any repairs yourself! Just ask me a few questions.