To our knowledge, there is no known general method 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 hard to understand, as the structure of most of them is unknown. In this article, by appealing to patch closure and using the lying over minimal prime technique, we overcome this obstacle and then obtain useful new results for computing the Zariski and flat closures of an infinite subset of the prime spectrum.
