The spectral spectrum of a unital commutative ring

Abstractly constructed prime spectra

We generalize one of the main results of abstract algebraic geometry, namely of the fact that the prime spectrum of a unital commutative ring is always a coherent topological space.In this generalization, which includes several other known ones, the role of ideals of is played by elements of an abstract complete lattice equipped with binary multiplication with for all.When no further conditions are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral.This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of we also make short additional remarks on semiprime elements.We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world.