= Ring structure of the p-adic integers
The <p-adic integers> form a <complete metric space> and a <discrete valuation ring>. Every nonzero $x$ is uniquely $p^k u$ with $k=v_p(x)\geq0$ and $u$ a <p-adic unit>. The prime $p$ is a <uniformizer>, and reduction modulo $p$ gives the <residue field> $\mathbb F_p$. The element $p$ has no inverse inside $\mathbb Z_p$, so the ring is not a <field>.
For a short ideal-classification outline, take a nonzero <ideal> and choose an element of smallest <P-adic valuation> $k$, which exists by the <well-ordering principle>. Its unit factor is invertible, so the ideal contains $p^k$. Every other element is divisible by $p^k$, proving that the ideal is $p^k\mathbb Z_p$. Completeness follows because $\mathbb Z_p$ is a <closed subset> of the <complete metric space> $\mathbb Q_p$. The zero ideal is separate from this nonzero-ideal classification.
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