I recently moved into a new apartment and am trying to fix a broken stainless steel sink basket. The old basket had a much longer thread(which broke) than the new one so I have to extend the pipe about three inches so that it will reach. The old pvc seems to have a bit of copper piping that was glued into it and has broken off on the inside of the pvc just near the top which won't allow me to slide a new copper extension in so I am trying to figure out how to create an extension. Can I saw off part of the existing pvc, add a joint and then an extension up to the basket or do I need to replace the whole piece? If it helps the pipe I am working with is the horizontal extension off the s-trap.

I'm having trouble figuring out what yo are trying to do. Can you post a picture? I think you are trying to repair a drain line that someone "rinky dinked" before you.

If it helps the pipe I am working with is the horizontal extension off the s-trap.
I don't wonder that Harold's confused. You start with a new basket strainer that the old tailpiece is too short.
You then jumo over to a piece of PVC that has a copper pipe glued into it. (that's a new one) I assume we're still talking about the short tailpiece. But then you muddy the waters by telling us it wasn't the drain tailpiece after all but the horizontal extension off the s-trap. Which not only confuses Harold but jut mystifies the hell out of me.
I) I still don't really know exactly what your complaint is.
2) "S" traps, (see image) don't have a horizontal extension.
Perhaps you could enlighten us with a better explanation and a picture.
Regards, Tom