The Pierce Spectrum of a Ring

A spectrum of a ring is a topological space associated with the ring which allows, up to an isomorphism, a representation of the ring as a ring of continuous functions on that spectrum. Many different spectra of a ring can be defined, each of them allowing a different representation of the ring, with particular properties. We are interested here in the Pierce spectrum of the ring. The idempotent elements of a commutative ring with unit constitute a Boolean algebra and the profinite space, corresponding to that Boolean algebra via the Stone duality, is the Pierce spectrum of the ring. The properties of this profinite Pierce spectrum, in the case of algebras over a ring, will play an essential role in developing the Galois theory of rings. When the ring is a field K, its spectrum is just a singleton and thus is never considered. Nevertheless, the spectra of the algebras over the field K were implicitly present in the Galois theory of fields.