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.

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).

scholarly journals Simple modules over Auslander regular rings

2017 ◽  
Vol 15 (1) ◽  
pp. 1618-1622
Author(s):  
Chonghui Huang ◽  
Lijing Zheng
Keyword(s):  
Projective Dimension ◽  
Global Dimension ◽  
Simple Modules ◽  
Equivalent Characterization ◽  
The Right ◽  
Regular Rings

AbstractIn this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of$\begin{array}{} \text{Ext}_{{\it\Lambda}}^{1}(S,\ {\it\Lambda}) \end{array} $is equal to 1 for any simpleΛ-moduleSwith grS= 1. As a result, we give some equivalent characterization of diagonal Auslander regular rings.

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

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.

scholarly journals THE HOMOGENISED ENVELOPING ALGEBRA OF THE LIE ALGEBRA sℓ(2,ℂ)

2014 ◽  
Vol 56 (3) ◽  
pp. 551-568 ◽  
Author(s):  
ROBERTO MARTINEZ-VILLA
Keyword(s):  
Lie Algebra ◽  
Projective Dimension ◽  
Global Dimension ◽  
Verma Modules ◽  
Regular Component ◽  
Koszul Duality ◽  
Koszul Algebra ◽  
Yoneda Algebra ◽  
Enveloping Algebra ◽  
Wild Algebra

AbstractIn this paper, we study the homogenised algebra B of the enveloping algebra U of the Lie algebra sℓ(2,ℂ). We look first to connections between the category of graded left B-modules and the category of U-modules, then we prove B is Koszul and Artin–Schelter regular of global dimension four, hence its Yoneda algebra B! is self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised Verma modules, which correspond to the lifting to B of the usual Verma modules over U, and prove that such modules are Koszul of projective dimension two. It was proved in Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these are located at the mouth of a regular component. In this way we obtain a family of components over a wild algebra indexed by ℂ.

scholarly journals The homological dimensions of simple modules

1993 ◽  
Vol 48 (2) ◽  
pp. 265-274 ◽  
Author(s):  
Nanqing Ding ◽  
Jianlong Chen
Keyword(s):  
Global Dimension ◽  
Simple Modules ◽  
Homological Dimensions ◽  
Coherent Ring ◽  
The Right

We prove that (a) if R is a commutative coherent ring, the weak global dimension of R equals the supremum of the flat (or (FP–)injective) dimensions of the simple R-modules; (b) if R is right semi-artinian, the weak (respectively, the right) global dimension of R equals the supremum of the flat (respectively, projective) dimensions of the simple right R-modules; (c) if R is right semi-artinian and right coherent, the weak global dimension of R equals the supremum of the FP-injective dimensions of the simple right R-modules.

scholarly journals Relative Gorenstein Dimensions over Triangular Matrix Rings

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2676
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
Juan Ramón García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Projective Dimension ◽  
Global Dimension ◽  
Projective Modules ◽  
Matrix Rings ◽  
Weakly Compatible ◽  
Gorenstein Algebra ◽  
Finite Projective Dimension ◽  
Triangular Matrix Ring

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

scholarly journals TAX EXCHANGE OF INFORMATION AND INTERNATIONAL COOPERATION IN BRAZIL

2015 ◽  
Vol 11 (1) ◽  
pp. 13-35 ◽  
Author(s):  
Heloisa Estellita ◽  
Frederico Silva Bastos
Keyword(s):  
International Cooperation ◽  
Tax Evasion ◽  
Legal Framework ◽  
Global Dimension ◽  
International Exchange ◽  
Tax Administration ◽  
International Treaties ◽  
Exchange Of Information ◽  
The Right ◽  
Supreme Court Rulings

Globalization and internationalization of companies are phenomena that need to be considered by modern tax administrations. In many situations, such as tax evasion, harmful tax competition and money laundering, domestic statutes seem to be ineffectual in a global dimension. Fo cope with that, new forms of regulation and regulators emerge. Under this view, an effort towards signing international treaties, conventions and agreements seems to be a feasible solution. The brazilian legal framework contains principles and rules that make international cooperation and exchange of information (eoi) with other countries possible. Furthermore, the brazilian tax administration has wide-ranging access powers to obtain information for international exchange purposes and has the tools to coercively produce such information. Brazil is following the right path to implement international exchange of information standards. However, there are some obstacles that need to be fixed for a more efficient implementation of these mechanisms. This article examines some topics of the brazilian legal and institutional framework on the tax exchange of information, such as a new model of approach of the tax administration, the tax transparency agenda and the international agreements on eoi matters, the brazilian supreme court rulings under bank secrecy and the rights of brazilian taxpayers regarding eoi.

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 Puberty and its Significance as Perceived by Prepubescents in Croatia

2021 ◽  
Vol 15 (2) ◽  
pp. 89-110
Author(s):  
Miluše Rašková ◽  
Dominika Provázková Stolinská ◽  
Michaela Bartošová
Keyword(s):  
Primary School ◽  
Sexuality Education ◽  
Human Life ◽  
Relevant Information ◽  
Global Dimension ◽  
Physical Growth ◽  
Reproductive Capacity ◽  
Test Items ◽  
Level Of Knowledge ◽  
The Right

Prepubescent children need to be prepared for puberty in an appropriate manner including all related associations and contexts. Children should learn the required knowledge before its onset—when they are in primary school. Knowledge is gained in the process of learning and represents the level of awareness. The cognitive level of knowledge represents the amount and quality of relevant information. Adolescence is a broadly defined stage of life, in which the reproductive capacity culminates and physical growth is completed. In the period of adolescence, psychological and social changes take place along with biological changes. Puberty is a significant element of sexuality education in the global dimension. The objective of the present educational research study is to identify the level of knowledge concerning the definition and significance of puberty among primary school pupils in Croatia and in the context of other selected countries (Czech Republic, Sweden, China, Spain). The data were collected by means of a knowledge achievement test. The content of the nine test items focused on the following: concept of puberty; definition of puberty; puberty age range; knowledge about physical changes in boys and girls; knowledge about other changes that puberty induces; significance of puberty in human life. To identify any statistically significant differences in the pupils’ responses by countries, the non-parametric Kruskal-Wallis test was used. The results suggest that prepubescents do not have the right knowledge about puberty and do not consider relevant associations and contexts in a comprehensive way. The incomplete knowledge suggests that prepubescents do not think about the significance of puberty in a comprehensive perspective but rather in various combinations of the biological, psychological and social areas.

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