In the book Algebraic Geometry by Hartshorne, page 74, it is said that the spectrum of a discrete valuation ring $R$ has only two points. How to show that a discrete valuation ring has only two prime ideals? In , it is said that a discrete valuation ring has exactly one non-zero maximal ideal.
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$\begingroup$Here is an argument starting from the definition of a DVR as a ring admitting a discrete valuation:
if $P$ is a non-zero prime ideal, then it contains a non-zero element, which can be written in the form $u t^n$, where $u$ is a unit (i.e. has valuation $0$), and $t$ is a uniformizer (i.e. has valuation $1$). Then $t^n \in P$ (multiply by $u^{-1}$), and then $t \in P$ (using primality). Thus the unique non-zero ideal of $R$ is the (maximal) ideal of elements of valuation $\geq 1$.
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