Complete Local Rings

Gert-Martin Greuel () and Gerhard Pfister ()
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Gert-Martin Greuel: University of Kaiserslautern, Department of Mathematics
Gerhard Pfister: University of Kaiserslautern, Department of Mathematics

Chapter 6 in A Singular Introduction to Commutative Algebra, 2002, pp 313-334 from Springer

Abstract: Abstract For certain applications the local rings K[x]〈x〉, x = (x1, ... , x n ), are not “sufficiently local”. As explained in Appendix A, Sections A.8 and A.9, the latter rings contain informations about arbitrary small Zariski neighbourhoods of 0 ∈ K n . Such neighbourhoods turn out to be still quite large, for instance, if n = 1 then they consist of K minus a finite number of points. If we are working over the field K = ℂ, respectively K = ℝ, we can use the convergent power series ring K{x} which contains information about arbitrary small Euclidean neighbourhoods of 0, and this is what we are usually interested in. For arbitrary fields, however, we have to consider the formal power series ring K[[x]] instead.

Keywords: Prime Ideal; Local Ring; Maximal Ideal; Standard Basis; Cauchy Sequence (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-3-662-04963-1_6

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