During replacing a bathtub, the drain pipe got caugt on the tub and it broke the PCV joint near the slab. The coupling broke, but the PCV piece that comes in through the slab fully intact. However, it is still glued into half of the coupling. There does not appear to be enough room to cut and simply put a new coupling on (although if that is the only way, then it may be possible). Is there some way to cut/heat/?? The coupling off, leaving the PVC piece in tact.

' Is there some way to cut/heat/?? The coupling off, leaving the PVC piece intact?"
No, when PVC is glued the two pieces melt into each other and form a weld.
You don't mention a square hole in the slab called a tub dap-out or tub hole. The trap sets down inside. Do you have such a hole in your slab? Tom

Best bet is to chip away the concrete around the pipe until you expose enough to cut the old coupler off and install a new one. Most slabs will chip out easily with a cold chisel.