One-Dimensional Semilocal Cohen-Macaulay Rings

Abstract

The local ring of a point on a curve is a one-dimensional local Cohen-Macaulay ring; in this chapter we study this class of rings. After proving some results on transversal elements in section 1, our main interest in section 2 is the integral closure of a one-dimensional local Cohen-Macaulay ring; we use Manis valuations in describing the integral closure. In section 3 we give necessary and sufficient conditions in order to ensure that the completion of a one-dimensional local Cohen-Macaulay ring which is a domain (resp. has no nilpotent elements) again is a domain (resp. has no nilpotent elements). Here the reader is supposed to be acquainted with the notion of the completion of a local ring and its properties.KeywordsPrime IdealLocal RingMaximal IdealValuation RingRegular ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.