where are ribosomes synthesized in a cell/

Ribosomes are synthesised in the nucleolus of nucleus.

Ribosome is made up of rRNA and protein. The rRNA part is synthesized from the DNA in the nucleus, particularly in a region called nucleolus. The protein parts come from another DNA message that specify the production of these proteins, translated to mRNA and then the mRNA is transported out of nucleus. It is then translated into protein in cytoplasm. The protein parts are transported back to nucleus and together with the rRNA produced in nucleolus, they are assembled into complete ribosome.