Spectrum is T0

This article gives a fact about the relation between ring-theoretic assumptions about a commutative unital ring and topological consequences for the spectrum
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Statement

Verbal statement

The spectrum of a commutative unital ring is a T0 space: given any two points, it cannot happen that each one is in the closure of the other one.

Proof

The proof follows directly from the following fact: given two prime ideals, such that each is contained in the other, the ideals must be equal.