The synthesis of postsynaptic cell receptor proteins primarily occurs in d) the rough endoplasmic reticulum.
The rough endoplasmic reticulum (RER) is studded with ribosomes on its cytoplasmic surface, which are the sites of protein synthesis. Once synthesized, these proteins are packaged into vesicles and transported to the Golgi apparatus for further processing, modification, and eventual delivery to their functional sites, such as the postsynaptic membrane in neurons.
Other options, like the Golgi apparatus and lysosomes, are involved in processing and sorting proteins but do not primarily synthesize them. The cytoplasm does play a role in various cellular activities, but for receptor protein synthesis, the rough endoplasmic reticulum is the correct answer.