Is the Caratheodory $\sigma$-algebra generated by a semi-ring $R$ only dependent on $\sigma(R)$?

Let $$\mu$$ be a measure defined on a certain $$\sigma$$-algebra $$\Sigma$$, and let $$R,Q\subseteq \Sigma$$ be two different semi-rings of sets within it, such that their generated $$\sigma$$-algebras coincide. That is: $$\sigma(R)=\sigma(Q)$$

If we restrict $$\mu$$ to these semi-rings we will get pre-measures on them, which we can then extend via Caratheodory. Will the two $$\sigma$$-algebras of Caratheodory measurable sets generated by these pre-measures necessarily be the same $$\sigma$$-algebra?

Phrased another way, can we take a pre-measure, extend it, then restrict it to a new but "just as large" semi-ring such that after re-extending it we'll end up with different measurable sets?