scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

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 TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS

2013 ◽  
Vol 94 (1) ◽  
pp. 133-144
Author(s):  
ZHAOYONG HUANG ◽  
XIAOJIN ZHANG
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Tilted Algebra ◽  
Gorenstein Algebras

AbstractLet $\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda $ admits a trivial maximal 1-orthogonal subcategory of $\text{mod } \Lambda $, then, for any indecomposable module $M\in \text{mod } \Lambda $, the projective dimension of $M$ is equal to one if and only if its injective dimension is also equal to one, and $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $\Lambda $ is a tilted algebra.

Homological Dimensions with Respect to a Semidualizing Module and Tensor Products of Algebras

2015 ◽  
Vol 22 (02) ◽  
pp. 215-222
Author(s):  
Maryam Salimi ◽  
Elham Tavasoli ◽  
Siamak Yassemi
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Tensor Products ◽  
Projective Dimension ◽  
Injective Dimension ◽  
Semidualizing Module ◽  
Homological Dimensions

Let C be a semidualizing module for a commutative ring R. It is shown that the [Formula: see text]-injective dimension has the ability to detect the regularity of R as well as the [Formula: see text]-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that id R(D) = n < ∞, then [Formula: see text] for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then [Formula: see text]. This generalizes the result of Enochs and Holm.

scholarly journals Some properties of non-commutative regular graded rings

1992 ◽  
Vol 34 (3) ◽  
pp. 277-300 ◽  
Author(s):  
Thierry Levasseur
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Supplementary Condition ◽  
Graded Rings ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Commutative Case ◽  
Finite Global Dimension ◽  
Image Position ◽  
Dimension One

Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a supplementary condition. Before stating it, we recall that the grade of a finitely generated (left or right) module is defined by

scholarly journals The finiteness of the Gorenstein dimension for Artin algebras

2019 ◽  
Vol 18 (06) ◽  
pp. 1950112
Author(s):  
René Marczinzik
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Artin Algebra ◽  
Artinian Rings ◽  
Gorenstein Dimension ◽  
Finite Global Dimension ◽  
Artin Algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

scholarly journals Behavior of the Auslander condition with respect to regradings

2019 ◽  
Vol 18 (09) ◽  
pp. 1950168 ◽  
Author(s):  
G.-S. Zhou ◽  
Y. Shen ◽  
D.-M. Lu
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Noetherian Ring ◽  
Invertible Element ◽  
Quotient Ring ◽  
Global Dimension ◽  
Injective Dimension ◽  
Graded Ring ◽  
Ideal Formula ◽  
The Ideal

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring [Formula: see text], its quotient ring modulo the ideal [Formula: see text] and its localization ring with respect to the Ore set [Formula: see text], where [Formula: see text] is a homogeneous regular normal non-invertible element of [Formula: see text].

Generalized Discrete Valuation Rings

1969 ◽  
Vol 21 ◽  
pp. 1404-1408 ◽  
Author(s):  
H.-H. Brungs
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Unit Element ◽  
Global Dimension ◽  
Order Type ◽  
Unique Factorization ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
The Right ◽  
Satisfy Property

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).In the following, R will always denote a ring with unit element satisfying (W).

scholarly journals On Presented Dimensions of Modules and Rings

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Dexu Zhou ◽  
Zhiwei Gong
Keyword(s):  
Projective Dimension ◽  
Global Dimension ◽  
Finite Presentation ◽  
The Right

We define the presented dimensions for modules and rings to measure how far away a module is from having an infinite finite presentation and develop ways to compute the projective dimension of a module with a finite presented dimension and the right global dimension of a ring. We also make a comparison of the right global dimension, the weak global dimension, and the presented dimension and divide rings into four classes according to these dimensions.

Relative Gorenstein global dimension

2016 ◽  
Vol 26 (08) ◽  
pp. 1597-1615 ◽  
Author(s):  
Driss Bennis ◽  
J. R. García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Upper Bound ◽  
Projective Dimension ◽  
Global Dimension ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Tilting Module ◽  
Injective Dimension ◽  
Wakamatsu Tilting Module ◽  
Bass Class ◽  
Gorenstein Injective

We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module [Formula: see text]. We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add[Formula: see text] and the [Formula: see text]-projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of [Formula: see text] is always bounded above by the [Formula: see text]-projective global dimension of [Formula: see text], these two dimensions being equal when the class of all [Formula: see text]-Gorenstein projective [Formula: see text]-modules is contained in the Bass class of [Formula: see text] relative to [Formula: see text]. Of course we also give the dual results concerning the relative Gorenstein injective global dimension.

scholarly journals Localizations of injective modules

1985 ◽  
Vol 28 (3) ◽  
pp. 289-299 ◽  
Author(s):  
K. R. Goodearl ◽  
D. A. Jordan
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Simple Module ◽  
Global Dimension ◽  
Injective Hull ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Noetherian Domain ◽  
Left And Right

The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

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