Do you know an example of a local ring? Another one? What about completions?
Solution
Standard examples of local rings include:
- any field, with maximal ideal \((0)\);
- a localization \(R_{\mathfrak m}\) at a maximal ideal, whose maximal ideal is \(\mathfrak mR_{\mathfrak m}\);
- a discrete valuation ring such as \(\mathbb Z_{(p)}\) or \(k[[t]]\);
- the formal power-series ring \(k[[x_1,\dots,x_n]]\), whose maximal ideal is \((x_1,\dots,x_n)\).
If \((R,\mathfrak m)\) is local, its \(\mathfrak m\)-adic completion \[ \widehat R=\varprojlim R/\mathfrak m^n \] is again local, with maximal ideal the kernel of \(\widehat R\to R/\mathfrak m\). For example, \[ \mathbb Z_p=\varprojlim_n\mathbb Z/p^n\mathbb Z \] is the \(p\)-adic completion of \(\mathbb Z_{(p)}\) and is local with maximal ideal \(p\mathbb Z_p\).