The Krull dimension theory of commutative Noetherian rings

2006 ◽  
pp. 237-245
Author(s):  
Louis Rowen
Keyword(s):  
Krull Dimension ◽  
Noetherian Rings ◽  
Dimension Theory

Torsion theoretic points and spaces

1984 ◽  
Vol 96 (3-4) ◽  
pp. 345-361 ◽  
Author(s):  
Harold Simmons
Keyword(s):  
Topological Space ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Decomposition Theory ◽  
Torsion Theories ◽  
Natural Decomposition

SynopsisFor each ring R, we construct a topological space pt (R) which includes as a subspace both the classical spectrum specR and the torsion theoretic spectrum R-sp. For many rings (e.g. rings with Krull dimension), spec R is a retract of pt (R) and the retraction map θ generalizes the Gabriel correspondence for noetherian rings. There is a natural decomposition theory on MOD-R which extends the Goldman theory in the same way that the tertiary theory extends the primary theory. The map θ provides a direct comparison between this new decomposition theory and the tertiary theory. The space pt (R) is closely connected with the lattice of hereditary torsion theories on R, and for fully bounded (not necessarily noetherian) R, this connection is very tight.

Krull Dimension of Noetherian Rings. The Principal Ideal Theorem

1987 ◽  
pp. 191-215
Author(s):  
Constantin Nǎstǎsescu ◽  
Freddy van Oystaeyen
Keyword(s):  
Principal Ideal ◽  
Krull Dimension ◽  
Noetherian Rings

scholarly journals Local dimension theory of tensor products of algebras over a ring

2018 ◽  
Vol 17 (06) ◽  
pp. 1850106
Author(s):  
Samir Bouchiba
Keyword(s):  
General Setting ◽  
Tensor Products ◽  
Krull Dimension ◽  
Dimension Theory ◽  
Boolean Ring ◽  
Local Dimension ◽  
Arbitrary Ring ◽  
Commutative Algebras ◽  
Local Properties ◽  
Nonzero Characteristic

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring [Formula: see text]. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field [Formula: see text] into the general setting of algebras over an arbitrary ring [Formula: see text]. For this sake, we introduce and study the notion of a fibered AF-ring over a ring [Formula: see text]. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings [Formula: see text] consisting of two [Formula: see text]-algebras [Formula: see text] and [Formula: see text] such that [Formula: see text], we introduce the inherent notion to [Formula: see text] of a [Formula: see text]-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product [Formula: see text]. As an application, we provide a formula for the Krull dimension of [Formula: see text] when either [Formula: see text] or [Formula: see text] is zero-dimensional as well as for the Krull dimension of [Formula: see text] when [Formula: see text] is a fibered AF-ring over the ring of integers [Formula: see text] with nonzero characteristic and [Formula: see text] is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of [Formula: see text] when [Formula: see text] is a Boolean ring. Actually, we prove that if [Formula: see text] and [Formula: see text] are rings such that [Formula: see text] is not trivial and [Formula: see text] is a Boolean ring, then dim[Formula: see text].

scholarly journals Some examples of modules over Noetherian rings

1982 ◽  
Vol 23 (1) ◽  
pp. 9-13 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Noetherian Ring ◽  
Essential Extension ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Finitely Generated

The purpose of this note is to prove the following result.Theorem 1. Let n be an integer greater than zero. There exists a prime Noetherian ring R of Krull dimension n + 1 and a finitely generated essential extension W of a simple R-module V suchthat(i) W has Krull dimension n, and(ii) W/V is n-critical and cannot be embedded in any of its proper submodules.We refer the reader to [6] for the definition and properties of Krull dimension.

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