Minimal and maximal spectra as the Stone-\v{C}ech compactification

In this paper, new advances on the compactifications of topological spaces, especially on the Stone-\v{C}ech and Alexandroff compactifications have been made. Among the main results, it is proved that the minimal spectrum of the direct product of a family of integral domains indexed by a set $X$ is the Stone-\v{C}ech compactification of the discrete space $X$. Dually, it is proved that the maximal spectrum of the direct product of a family of local rings indexed by $X$ is also the Stone-\v{C}ech compactification of the discrete space $X$. The Alexandroff (one-point) compactification of a discrete space is constructed by a new method. Next, we proceed to give a natural and quite simple way to construct ultra-rings. Then this new approach is used to obtain several new results on the Stone-\v{C}ech compactification.