Exercise A0.V.2.20

Proposition

Let $R$ be a DVR with valuation $v$. Then $R$ is a PID.

Proof

Let $t \in R$ be such that $v(t) = 1.$ Such $t$ necessarily exists by surjectivity of $v$ and $v(x) \geq 0$ construction as in 2.19.

Let $I$ be an ideal of $R.$ If $I = (0)$ or $I = R,$ then $I$ is principal and we are done. Suppose otherwise, that $I$ is a proper non-zero ideal.

Let $i \in I$ be a non-zero element minimizing $v.$ Let $n = v(i).$

Note that since $v(i) = v(t^n),$ $v(i/t^n) = 0$ and hence $i/t^n$ is a unit. Thus $(i) = (t^n).$

Let $a \in I$ be non-zero. Since $v(a) \geq n,$ there exists $a/i \in R.$ Hence $(a/i) \cdot i = a,$ thus $I = (i).$

$\blacksquare$