BMW didn't build them without oxygen sensors. Your car is probably running as weird as it is because it doesn't have one installed. The 316i (edit: which could be an M40, by the way -- BMW installed M40s in the 316i up until early 1994, even though some cars got M43s as early as Spring 1993) has one sensor, with the part number that mullethunter3 provided.
So, yes, if you install one, it will work and your car should run much better.