As stated in the title I am interested in examples of noetherian local integral domains $R$ of Krull dimension 1.
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2Unfortunately, the [DaRT search](http://ringtheory.herokuapp.com/commsearch/commresults/?has=5&has=4&has=2&lacks=3) for this does not include searching by Krull dimension at present (although I think Watson already requested it!) But as you can see from the three (at present) hits, I've been making an effort to include the Krull dimension in the notes. It turns out two of the three are $1$ dimensional: $k[[x^2, x^3]]$ and $k[[x]]$. The latter is a discrete valuation ring as in Francesco's answer, but the former is not. – rschwieb Jul 05 '17 at 13:35
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You may consider for instance discrete valuation rings. Archetipal examples of such rings are
 the local ring $\mathcal{O}_{X, \, x}$ of a smooth curve $x$ at a point $x \in X$;
 the ring $\mathbb{Z}_p$ of $p$adic integers (for any prime $p$);
 the ring $\mathbb{K}[[x]]$ of formal power series in one variable $x$ over some field $\mathbb{K}$.
Francesco Polizzi
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Ok thank you. I am not that educated in algebraic geometry so I am not aware of the notion of a smooth curve. Could you give some examples of schemes $X$ that are smooth curves. In the best case some affine ones. – Constantin K Jul 05 '17 at 12:46



any smooth affine curve of the form $f(x, y)=0$, where $f \in k[x,y]$. – Francesco Polizzi Jul 05 '17 at 12:59