In 1986, Jacobson has defined the Brauer ring B(E, D) for a finite Galois field extension E/D, whose unit group canonically contains the Brauer group of D. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group G is trivial, the Brauer ring of the trivial extension E/E becomes naturally isomorphic to the group ring of the Brauer group of E. In this paper, we generalize this result to any finite group G via the theory of the restriction functor, by means of the well-understood functor −+. More generally, we determine the structure of the F-Burnside ring for any additive functor F. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate F related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension ℂ/ℝ.