Some Remarks on One-Dimensional Local Domains | Zendy

Ryoichi Nagasawa | Zendy

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Some Remarks on One-Dimensional Local Domains

Author(s) -

Ryoichi Nagasawa

Publication year - 1975

Publication title -

publications of the research institute for mathematical sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.786

H-Index - 39

eISSN - 1663-4926

pISSN - 0034-5318

DOI - 10.2977/prims/1195191685

Subject(s) - mathematics , noetherian , quotient , local ring , integral domain , pure mathematics , regular local ring , multiplicity (mathematics) , ring (chemistry) , locality , extension (predicate logic) , quadratic equation , unique factorization domain , local field , local cohomology , singularity , field (mathematics) , ring of integers , closure (psychology) , domain (mathematical analysis) , algebra over a field , mathematical analysis , algebraic number field , geometry , law , factorization , finitely generated abelian group , philosophy , algorithm , linguistics , chemistry , computer science , political science , programming language , organic chemistry

Let R be a one-dimensional (noetherian) local domain with field of quotients Q. Then any ring extension S of R in Q is obtained as a ring of quotients of some integral extension C of R. Here, if S is local and if C can be chosen to be finite over R, then we call S an K-locality. If R is analytically ramified, then .R does not satisfy the finiteness condition for integral extensions in Q (cf. [3], p. 122, Exercise 1). In other words, .R possesses at least one latent singularity with respect to a certain analytic branch of R which can not be resolved by any quadratic dilatations. The purpose of this note is to give a necessary and sufficient condition for the finiteness of ring extensions S as .R-modules, and to prove a characterization of ^-localities by making use of the concept of latent multiplicity found in [L] and [4] (more detailed accounts of this theory may be found in [3]).

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