Spectral closures of an infinite subset of the prime spectrum

In the literature, there is no known general method (formula) to compute the Zariski closure of an ``infinite’’ subset of the prime spectrum. This problem indeed deals with the prime ideals of an infinite direct product of nonzero commutative rings that are very complicated to understand (the structure of most of them is unknown). In this article, by appealing to the patch closure and using laying over minimal prime technique, we overcome the above obstacle and then obtain new and quite useful results for computing the Zariski and flat closures of an infinite subset of the prime spectrum.