An isomorphism about the central localization of a monoid

Question

Let $M$ be a monoid and $K$ an algebraically closed field. Then we can get a semigroup algebra $KM$. Assume the element $z$ is a central and regular element of $KM$, when we consider the central localization $M\langle z\rangle^{-1}$, we have \begin{align} M\langle z\rangle^{-1}\cong M/(z=1)\times \langle z, z^{-1}\rangle. \end{align} My question is how to proof this isomorphism? Any answers will be appreciated.