Local Ring - Examples

Examples

• All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
• An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields.
• Every ring of formal power series over a field (even in several variables) is local; the maximal ideal consists of those power series without constant term.
• Similarly, the algebra of dual numbers over any field is local. More generally, if F is a field and n is a positive integer, then the quotient ring F/(Xn) is local with maximal ideal consisting of the classes of polynomials with zero constant term, since one can use a geometric series to invert all other polynomials modulo Xn. In these cases elements are either nilpotent or invertible.
• The ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator: this is the integers localized at 2.

More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization.