Goldie and Krull Dimensions of Rings and Modules

2016 ◽  
pp. 174-240
Keyword(s):  
Krull Dimensions

scholarly journals Classification of Simple Modules of the Ore Extension $$K[X][Y; f\frac{d}{dX}]$$

2019 ◽  
Vol 14 (2) ◽  
pp. 317-325
Author(s):  
V. V. Bavula
Keyword(s):  
Explicit Description ◽  
Ore Extension ◽  
Simple Modules ◽  
Krull Dimensions

Abstract For the algebras $$\Lambda $$Λ in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of $$\Lambda $$Λ are computed.

Krull dimensions of divisorially graded rings

1991 ◽  
Vol 19 (12) ◽  
pp. 3447-3464
Author(s):  
M. Jesus Asensio ◽  
Gomez Jose ◽  
Torrecillas Bias
Keyword(s):  
Graded Rings ◽  
Krull Dimensions

Down–up algebras over a polynomial base ring 𝕂[t1,…,tn]

2015 ◽  
Vol 15 (02) ◽  
pp. 1650024
Author(s):  
Xin Tang
Keyword(s):  
Prime Ring ◽  
Global Dimension ◽  
Noetherian Domain ◽  
Base Ring ◽  
Kirillov Dimension ◽  
Krull Dimensions

We study a class of down–up algebras 𝒜(α, β, ϕ) defined over a polynomial base ring 𝕂[t1,…,tn] and establish several analogous results. We first construct a 𝕂-basis for the algebra 𝒜(α, β, ϕ). As an application, we completely determine the center of 𝒜(α, β, ϕ) when char 𝕂 = 0, and prove that the Gelfand–Kirillov dimension of 𝒜(α, β, ϕ) is n + 3. Then, we prove that 𝒜(α, β, ϕ) is a noetherian domain if and only if β ≠ 0, and 𝒜(α, β, ϕ) is Auslander-regular when β ≠ 0. We show that the global dimension of 𝒜(α, β, ϕ) is n + 3, and 𝒜(α, β, ϕ) is a prime ring except when α = β = ϕ = 0. Finally, we obtain some results on the Krull dimensions, isomorphisms and automorphisms of 𝒜(α, β, ϕ).

scholarly journals Krull dimension and invertible ideals in Noetherian rings

1976 ◽  
Vol 20 (2) ◽  
pp. 81-86 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Cartesian Product ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Infinite Sequences ◽  
Krull Dimensions

In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.

scholarly journals Global and Krull Dimensions of Quantum Weyl Algebras

1999 ◽  
Vol 216 (2) ◽  
pp. 405-416 ◽  
Author(s):  
Hisaaki Fujita ◽  
Ellen Kirkman ◽  
James Kuzmanovich
Keyword(s):  
Weyl Algebras ◽  
Krull Dimensions

On the Krull Dimensions of the Steenrod Algebra

2012 ◽  
Vol 40 (10) ◽  
pp. 3859-3866 ◽  
Author(s):  
Robert P. Stephens
Keyword(s):  
Steenrod Algebra ◽  
Krull Dimensions
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