The piecewise Noetherian property in power series rings over a valuation domain

For an integral domain D of dimension n, the dimension of the polynomial ring D[ x ] is known to be bounded by n + 1 and 2n + 1. While n + 1 is a lower bound for the dimension of the power series ring D[[ x ]], it often happens that D[[ x ]] has infinite chains of primes. For example, such chains exist if D is either an almost Dedekind domain that is not Dedekind or a one-dimensional nondiscrete valuation domain. The main concern here is in developing a scheme by which such chains can be constructed in the gap between MV[[ x ]] and M[[ x ]] when V is a one-dimensional nondiscrete valuation domain with maximal ideal M. A consequence of these constructions is that there are chains of primes similar to the set of ω1 transfinite sequences of 0's and 1's ordered lexicographically.

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Prime t-ideals in power series rings over a discrete valuation domain

Journal of Algebra ◽

10.1016/j.jalgebra.2010.09.015 ◽

2010 ◽

Vol 324 (12) ◽

pp. 3401-3407 ◽

Cited By ~ 2

Author(s):

Mi Hee Park

Keyword(s):

Power Series ◽

Valuation Domain ◽

Discrete Valuation Domain ◽

Power Series Rings

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PrimeT-Ideals in Polynomial and Power Series Rings Over a Pseudo-Valuation Domain

Communications in Algebra ◽

10.1080/00927872.2014.897190 ◽

2014 ◽

Vol 43 (1) ◽

pp. 51-58

Author(s):

Mi Hee Park

Keyword(s):

Power Series ◽

Valuation Domain ◽

Power Series Rings

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Some results concerning the local analytic branches of an algebraic variety

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100028541 ◽

1953 ◽

Vol 49 (3) ◽

pp. 386-396 ◽

Cited By ~ 1

Author(s):

D. G. Northcott

Keyword(s):

Algebraic Geometry ◽

Power Series ◽

Algebraic Variety ◽

Recent Progress ◽

Rapid Development ◽

Noetherian Rings ◽

Local Rings ◽

Power Series Rings ◽

Analytic Algebra ◽

Topological Theories

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

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