2001 300M Alternator/Battery Question

I had a new battery recently installed in my '01 300M and a couple of weeks later, the alternator went bad. As it turns out, the new battery has an internal short that causes the voltage to drop when a load is applied. My question is, did the bad battery cause the alternator to burn out? Is it possible that the alternator's regulator tried to compensate for low battery voltage by running at full output for a couple of weeks, resulting in it going bad (that's my opinion, but I'd like to get some feedback)?

Thanks!