scholarly journals A class of maximal orders integral over their centres

1983 ◽  
Vol 24 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ring ◽  
Global Dimension ◽  
Local Rings ◽  
Maximal Orders ◽  
Finite Global Dimension ◽  
Krull Domain ◽  
Rank One

In a recent paper [1], Brown, Hajarnavis and MacEacharn have considered non-commutative Noetherian local rings of finite global dimension which are integral over their centres. For such a ring Rthey have shown:(i) R is a prime ring whose Krull and global dimensions coincide;(ii) R = ∩ RP where p runs through the set of rank one primes of the centre of R, and each Rp is hereditary;(iii) the centre of R is a Krull domain.

scholarly journals Some examples of noncommutative local rings

1990 ◽  
Vol 32 (1) ◽  
pp. 79-86 ◽  
Author(s):  
D. A. Jordan
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Global Dimension ◽  
Local Rings ◽  
General Area ◽  
Skew Polynomial Ring ◽  
Finite Global Dimension ◽  
Skew Polynomial

In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].

scholarly journals Direct sums of exact covers of complexes

2006 ◽  
Vol 98 (2) ◽  
pp. 161 ◽  
Author(s):  
Alina Iacob
Keyword(s):  
Global Dimension ◽  
Noetherian Rings ◽  
Local Rings ◽  
Direct Sums ◽  
Direct Sum ◽  
Finite Global Dimension ◽  
Flat Modules ◽  
Projective Resolutions ◽  
Finitely Generated Modules ◽  
The Given

A ring $R$ is left noetherian if and only if the direct sum of injective envelopes of any family of left $R$-modules is the injective envelope of the direct sum of the given family of modules (or equivalently, if and only if the direct sum of any family of injective left $R$-modules is also injective). This result of Bass ([2]) led to a series of similar closure questions concerning classes of modules and classes of envelopes and covers (Chase in [4] considers the question of the closure of the class of flat modules with respect to products). Motivated by Bass' result we consider the question of direct sums of exact covers of complexes. From the close connection between minimal injective resolutions of modules and exact covers of complexes it seemed reasonable to conjecture that we get this closure over left noetherian rings. In this paper we show that this is not the case and that under various additional hypotheses on the ring that in fact the ring must have finite left global dimension for this to happen. Our results raise what we consider an interesting question about characterizing the local rings of finite global dimension in terms of a certain property of minimal projective resolutions of finitely generated modules over the local ring. We also consider the closely related question of when the direct sum of DG-injective complexes is DG-injective.

scholarly journals The graded Cartan matrix and global dimension of 0-relations algebras

1987 ◽  
Vol 30 (3) ◽  
pp. 351-362 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Artinian Ring ◽  
Global Dimension ◽  
Cartan Matrix ◽  
Integral Matrix ◽  
Finite Global Dimension ◽  
Composition Factors

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

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 Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components

1998 ◽  
Vol 40 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Zygmunt Pogorzały
Keyword(s):  
Unit Element ◽  
Global Dimension ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Stable Category ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Stable Categories

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

2015 ◽  
Vol 67 (1) ◽  
pp. 28-54 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Razieh Vahed
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Derived Categories ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Finite Global Dimension

AbstractWe study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

scholarly journals Anick resolution and Koszul algebras of finite global dimension

2017 ◽  
Vol 45 (12) ◽  
pp. 5380-5383 ◽  
Author(s):  
Vladimir Dotsenko ◽  
Soutrik Roy Chowdhury
Keyword(s):  
Global Dimension ◽  
Koszul Algebras ◽  
Finite Global Dimension

Global dimension of graded local rings

1996 ◽  
Vol 24 (7) ◽  
pp. 2399-2405 ◽  
Author(s):  
Li Huishi
Keyword(s):  
Global Dimension ◽  
Local Rings
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