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Krull dimension and noetherianness

1976; Duke University Press; Volume: 20; Issue: 2 Linguagem: Inglês

10.1215/ijm/1256049903

1945-6581

Bharat Sarath,

Advanced Topics in Algebra

This paper essentially deals with the following question: Over what rings are all modules with Krull dimension (as defined in [1]; Gordon-Robson A.M.S. memoirs) noetherian.The problem originates with [-2] Theorem 4.2 where it is shown that k-rings of Krull-dimension are, among many other things, noetherian.In attempting to generalize this to higher dimensions it transpires that all modules with Krull dimension over a V-ring are noetherian and this leads to the question of other rings having this property.We tackle this mainly in Section 2 devoting Section to proving the following somewhat independent result about noetherian V-rings: a ring R is a noetherian V-ring if and only if every R-module M has a minimal generating set and given a submodule N of M every minimal generating set of N can be extended to a minimal generating set of M. In Section 2 we prove (Theorem 2.8) that over a ring R, every module with Krull-dimension is noetherian if and only if every non-noetherian module has a proper non-noetherian submodule.Constructing an analogue of we show that there are non-noetherian modules with Krull-dimension over a polynomial ring and also study the case of group rings.It has been pointed out to me that M. Teply has proved that V-rings with Krull dimension are noetherian using Proposition 1.5 and the resulting iso- morphism constructed in the proof of our Theorem 1.6, though the work is unpublished.I would also like to acknowledge my indebtedness to Dr. K. Varadarajan for his valuable advice and the many improvements he suggested.All the rings in this paper possess a unit and all modules are left unital.All properties will be assumed to be left properties e.g., "ideal" will mean "left ideal".The symbol ( ) will denote "module generated by by" i.e., (C) will mean the module generated by C, will denote set theoretic complement and {b} the singleton set consisting of b.1. Irredundant and redundant subsets of a module DEFINITION 1.1.Let M be a module, B a subset of M. We say B is ir- redundant iff A B, (A) (B) A B. If B is not irredundant we call it redundant.Remarks 1.2.(i) If B _ _ _ M is irredundant and A _ _ _ B then A is irredundant.(ii) If {B,},j is a family of irredundant subsets of M totally ordered by inclusion then , B, is irredundant.