The short answer is, “it depends on the business.” You can bet that Jeffrey Archer worked on his novels while in prison, but I’m pretty sure Bernie Madoff isn’t “investing” (or whatever he called it – what’s a verb for running a Ponzi Scheme?) anyone’s money, although one can never be too sure with some people.
It also depends on the crime for which you have been incarcerated, the prison, and the state’s laws, regulations and prison policies. It’s not inconceivable that if you invent, say, a better mousetrap, while incarcerated, you might be granted more leeway than if you invent a designer high-tech shiv (just sayin’), but whatever you do, please read up on the law, ask the prison authorities, and ask a lawyer.
Here in Oregon that means starting with the Oregon Revised Statutes (ORS) and the Oregon Administrative Rules (OAR), e.g. Chapter 291 Department of Corrections, perhaps including but not limited to 291-105-0015 Rules of Misconduct.
Of course one also needs to learn about running a business! There are a lot of places to start with that, and the Oregon blog is a great place to begin.