"After putting the RAM in empty Sockets OS was not able to detect the RAM, the issue seems to be due to limitation of 32 bit OS."
Did the system recognize the memory in the BIOS Setup (F2)? If so, then yes, 32-bit operating systems can only use 4GB of RAM. The "usable" amount will be less, as some of that is budgeted to manage hardware devices.
The Product Key is good for either 32-bit or 64-bit.