  In this paper, using commutative algebra, then new and significant advances
on the compactifications of topological spaces, especially on the
Stone-\v{C}ech and Alexandroff compactifications have been made. Especially
among them, it is proved that the minimal spectrum of the direct product of a
family of integral domains and also the maximal spectrum of the direct product
of a family of local rings both indexed by a set $X$ are the Stone-\v{C}ech
compactifications of the discrete space $X$. Some applications of this study
are given. Especially, it is shown that the Stone-\v{C}ech compactification of
an arbitrary topological space $X$ is obtained from the Stone-\v{C}ech
compactification of the discrete space $X$ by passing to a its certain
quotient. A natural and simple way to construct ultra-rings is introduced, then
this new approach is used to obtain nontrivial results on the Stone-\v{C}ech
compactification.
