I'm sure you already know this, but for your benefit and possibly that of others, you have to be careful w/what you want to do. Since your 'home' is a rented unit, you need the Landlord's permission to do any 'changes' to the apartment. (If anyone wants to know the consequences, ask any Landlord, preferably NOT yours.)
Anyways, unless you ADD something in between the walls, you're gonna hear those doors closing. I don't think an insulation board is a good idea because it changes the 'profile' of anything that it comes in contact with. This means more finishing work that needs to be done after you add the board. PROBABLY the best thing to do is to talk to the Landlord about ripping down the walls, then adding sound-defening material, then re-doing the walls. It's costly but should be enough to deafen the noise to a 'thud'. For me (and I'm only speaking for me), I'd look for another place to live. I'm not aware of any 'special circumstances' that are keeping you there but if they're THAT good, then do the job the RIGHT way. If you do move out, be sure you are SURE of what's between you and the other unit(s). Avoid taking a Landlord's word for it. Find a way of proving their word. (Sometimes, you can pull a Heat Vent or something and look inside.)
My best to ya and hope this helps.
Jay J -Moderator
PS: Search the Main Forum Page for the keyword: sound That may give you other ideas. Your question is certainly NOT the first one that's been asked ...