I know that for processes to exist, in the following entropy equation
$dS/dt = {\dot Q}/T + {\dot\sigma}$
${\dot\sigma}$ has to be equal or bigger than zero. The thing is that I also saw a version in which ${\sigma}$ had to be equal or bigger than zero. But even if ${\sigma}$ < 0, ${\dot\sigma}$ = 0 . This is what confuses me, which assumption is the right one, the one regarding ${\dot\sigma}$ or ${\sigma}$?
You are aware that the cumulative amount of entropy generated in a system by a process $\sigma$ is related to the instantaneous rate of entropy generation $\dot{\sigma}$ by the equation $\sigma(t)=\int_0^t{\dot{\sigma(t')}dt'}$ (where t' is a dummy variable of integration), right? So, in an irreversible process both $\dot{\sigma}$ and $\sigma$ must always be positive, and, in a reversible process they both must always be zero.
Incidentally, in the equation you wrote, the T is supposed to be the temperature at the system boundary where the heat transfer $\dot{Q}$ is occurring (not the average system temperature). For a reversible process, the two temperatures are equal, but, in an irreversible process, they are not.
