Stone type representation theorems and dualities by power set ring.

In this paper, it is shown that the Boolean ring of a commutative ring is isomorphic to the clopen ring of its prime spectrum. The topological version of this result is also proved which states that the space of connected components of a compact space is homeomorphic to the prime spectrum of its clopens. In particular, Stone's Representation Theorem is generalized from Boolean rings to arbitrary commutative rings and also the Stone duality is easily deduced. The prime spectrum of the Boolean ring of a given ring $R$ is identified with the Pierce spectrum of $R$. Then as an application, it is shown that the prime spectrum of a ring $R$ is discrete iff the Boolean ring of $R$ is isomorphic to the power set ring of its prime spectrum. As another major result, it is shown that a morphism of rings between complete Boolean rings preserves suprema iff the induced map between the corresponding prime spectra is an open map. This result leads us to a Stone type duality which states that the category of complete Boolean rings is the dual of the category of compact extremally disconnected spaces. This duality in particular yields that the injective objects of the category of Boolean rings are precisely the complete Boolean rings. New characterizations for the completeness of Boolean rings are also given. Finally, general results in the fixed-point theory have been obtained.