How can prokaryotes live without a nucleus?
Although prokaryotes do not have a nucleus (or other membrane-bound organelles), the do still have DNA. The DNA is a single loop, in an area of the cell called the nucleoid region (see image).
Prokaryotic DNA acts in a very similar way to eukaryotic DNA. It provides a template for mRNA synthesis (transcription), leading to protein synthesis at the ribosome (translation).
To reproduce the cell, the DNA loop is replicated, and one copy moves to each side of the cell as part of binary fission.