Does a company have the right to terminate an employee with no warning,write up, and there is no rule violation?

It depends but if you are at-will, yes.  Unless there are mitigating factors or discrimination issues in play, you will have a tough time pursuing the employer.  Also, if you have a contract, of course there are different rules.  Finally you should double check with the handbook if there is one.  Companies unfortunately are allowed to contract their work force.