I know that if $\mathsf C$ is a category and $\Sigma$ is some class with a right calculus of fractions then the localization functor $\varphi: \mathsf C\longrightarrow \mathsf C[\Sigma ^-1]$ is right exact.
I'm pretty sure the dual says that if $\Sigma$ also has a left calculus of fractions then $\varphi$ is left exact.
For commutative rings, I know localization is an exact functor.
What about the noncommutative case? Is localization functorial, and is this functor exact?
Also, why, intuitively, does a calculus of fractions make localization half-exact?