A New Construction of the Injective Hull

1968 ◽  
Vol 11 (1) ◽  
pp. 19-21 ◽  
Author(s):  
Isidore Fleischer
Keyword(s):  
Essential Extension ◽  
Injective Hull ◽  
Minimal Cardinality ◽  
New Construction ◽  
Definition Of

The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.

scholarly journals Models and scales for quality control: toward the definition of specifications (GOA-LOG) for the generation and re-use of HBIM object libraries in a Common Data Environment

2021 ◽  
Author(s):  
Raffaella Brumana ◽  
Chiara Stanga ◽  
Fabrizio Banfi
Keyword(s):  
Model Generation ◽  
Model Concept ◽  
Generative Process ◽  
New Construction ◽  
Phases Of Development ◽  
Data Environment ◽  
Definition Of ◽  
Object Models ◽  
Geometric Content ◽  
New Buildings

AbstractThe paper focuses on new opportunities of knowledge sharing, and comparison, thanks to the circulation and re-use of heritage HBIM models by means of Object Libraries within a Common Data Environment (CDE) and remotely-accessible Geospatial Virtual Hubs (GVH). HBIM requires a transparent controlled quality process in the model generation and its management to avoid misuses of such models once available in the cloud, freeing themselves from object libraries oriented to new buildings. The model concept in the BIM construction process is intended to be progressively enriched with details defined by the Level of Geometry (LOG) while crossing the different phases of development (LOD), from the pre-design to the scheduled maintenance during the long life cycle of buildings and management (LLCM). In this context, the digitization process—from the data acquisition until the informative models (scan-to-HBIM method)—requires adapting the definition of LOGs to the different phases characterizing the heritage preservation and management, reversing the new construction logic based on simple-to-complex informative models. Accordingly, a deeper understanding of the geometry and state of the art (as-found) should take into account the complexity and uniqueness of the elements composing the architectural heritage since the starting phases of the analysis, adopting coherent object modeling that can be simplified for different purposes as in the construction site and management over time. For those reasons, the study intends (i) to apply the well-known concept of scale to the object model generation, defining different Grades of Accuracy (GOA) related to the scales (ii) to start fixing sustainable roles to guarantee a free choice by the operators in the generation of object models, and (iii) to validate the model generative process with a transparent communication of indicators to describe the richness in terms of precision and accuracy of the geometric content here declined for masonry walls and vaults, and (iv) to identifies requirements for reliable Object Libraries.

Injectivity in Equational Classes of Algebras

1972 ◽  
Vol 24 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Alan Day
Keyword(s):  
Abelian Groups ◽  
Essential Extension ◽  
Boolean Algebras ◽  
Equational Class ◽  
Injective Hull ◽  
Fundamental Notion ◽  
First Results ◽  
Initial Results

The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.

The zoom-on-possessee construction in Kam (Dong): the anatomy of a new construction type

2005 ◽  
Vol 41 (2) ◽  
pp. 307-352 ◽  
Author(s):  
MATTHIAS GERNER
Keyword(s):  
Word Order ◽  
Body Part ◽  
Guizhou Province ◽  
General Definition ◽  
Republic Of China ◽  
Noun Incorporation ◽  
New Construction ◽  
Construction Type ◽  
Definition Of ◽  
Asian Languages

Kam, a Kadai language spoken in Guizhou province (People's Republic of China), has a family of intransitive possessive constructions with the word order ‘Possessor–Verb–Possessee’. (The basic word order in Kam is SV and AVO.) While two recent papers have featured this unique construction type for an array of other Southeast Asian languages, they fail to acknowledge its distinct semantic value in contrast to the related construction type ‘Possessee–Possessor–Verb’. The former construction type displays a so-called ‘zoom-effect’: the possessor is predicated IN, AT or THROUGH his/her/its possessee; the predication zooms from the possessor on his/her/its possessee. The latter construction, in contrast, views the possessee as an entity separated from its possessor, and the predicate as applying solely to the possessee. After illustrating the ‘zoom-effect’ for a representative sample of Kam constructions, I demonstrate that ‘zoom-effects’ do not merely exist when the possessee–possessor compound has the zero-role (=intransitive subject) as above, but also when it assumes other semantic roles (e.g. patient, force, etc.). A general definition of this construction type, called ‘zoom-on-possessee construction’, is proposed; it enables us to unify and account for an array of hitherto disparate construction types that run in the literature under labels such as ‘proprioceptive state expressions’, ‘body part locative constructions’, ‘dative of affect’, etc. Furthermore, I discuss in some detail whether zoom-on-possessee constructions are better accounted for within a multi-stratal or a mono-stratal framework and, finally, whether the concept of noun-incorporation has any relevance.

scholarly journals Modules Whose Cyclic Submodules Have Finite Dimension

1976 ◽  
Vol 19 (1) ◽  
pp. 1-6 ◽  
Author(s):  
David Berry
Keyword(s):  
Associative Ring ◽  
Finite Dimension ◽  
Essential Extension ◽  
Injective Hull ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Unitary Right

R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.

Use of BIM in rehabilitation and assessment of the built heritage: from the visible to the intangible

2019 ◽  
Author(s):  
Hélder S. Sousa ◽  
Carmen Sguazzo ◽  
Manuel Cabaleiro
Keyword(s):  
Relevant Information ◽  
Destructive Testing ◽  
Building Information Modelling ◽  
Information Modelling ◽  
Traditional Assessment ◽  
Existing Structures ◽  
Built Heritage ◽  
Building Information ◽  
New Construction ◽  
Definition Of

<p>Building Information Modelling (BIM) has been increasingly expanding its application to different fields of civil engineering and Historic building information modelling (HBIM) is an example of that. Although, the concept has already drawn the attention of several researchers, there are still many limitations to a full and holistic process that may take HBIM to the same level of applicability that BIM used for new construction has.</p><p>Traditionally, assessment of existing structures, specially heritage structures, begin with the documentation of all important information dealing with the history, characteristics, type, material, uses and applied techniques, among other relevant information that may be retrieved by different sources. Further on, a geometrical survey accompanied with visual inspection and non or semi destructive testing leads to the geometry definition of the structure and to its condition (damage/defects) mapping. All of this information, must be analysed for consequent structural assessment and after stored in a proper database in order to monitor the condition change of the structure along time.</p><p>This paper, presents a framework for use of BIM in rehabilitation and assessment of the built heritage, based on the review of recent works, as to allow a better understanding of the potential for the management of important and significant structures. The paper deals with the dilemma of bringing what a “traditional” assessment can see to how intangible information may be applied.</p>

scholarly journals On τ-completely decomposable modules

2004 ◽  
Vol 70 (1) ◽  
pp. 163-175 ◽  
Author(s):  
Septimiu Crivei
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Torsion Theory ◽  
Positive Answer ◽  
Essential Extension ◽  
Injective Module ◽  
Injective Hull ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Torsion Theories

For a hereditary torsion theory τ, a moduleAis called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ringRthat is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.

Strongly s-dense injective hull and Banaschewski’s theorems for acts

2020 ◽  
Vol 70 (2) ◽  
pp. 251-258
Author(s):  
Hasan Barzegar
Keyword(s):  
Sufficient Conditions ◽  
Essential Extension ◽  
Explicit Description ◽  
Injective Hull ◽  
Absolute Retract ◽  
Different Types ◽  
Important Notion

Abstract For a class 𝓜 of monomorphisms of a category, mathematicians usually use different types of essentiality. Essentiality is an important notion closely related to injectivity. Banaschewski defines and gives sufficient conditions on a category 𝓐 and a subclass 𝓜 of its monomorphisms under which 𝓜-injectivity well-behaves with respect to the notions such as 𝓜-absolute retract and 𝓜-essentialness. In this paper, 𝓐 is taken to be the category of acts over a semigroup S and 𝓜sd to be the class of strongly s-dense monomorphisms. We study essentiality with respect to strongly s-dense monomorphisms of acts. Depending on a class 𝓜 of morphisms of a category 𝓐, In some literatures, three different types of essentialness are considered. Each has its own benefits in regards with the behavior of 𝓜-injectivity. We will show that these three different definitions of essentiality with respect to the class of strongly s-dense monomorphisms are equivalent. Also, the existence and the explicit description of a strongly s-dense injective hull for any given act which is equivalent to the maximal such essential extension and minimal strongly s-dense injective extension with respect to strongly s-dense monomorphism is investigated. At last we conclude that strongly s-dense injectivity well behaves in the category Act-S.

On infinite direct sums of lifting modules

2017 ◽  
Vol 10 (03) ◽  
pp. 1750049
Author(s):  
M. Tamer Koşan ◽  
Truong Cong Quynh
Keyword(s):  
Present Article ◽  
Essential Extension ◽  
Injective Hull ◽  
Direct Sums ◽  
Simple Modules ◽  
Direct Sum

The aim of the present article is to investigate the structure of rings [Formula: see text] satisfying the condition: for any family [Formula: see text] of simple right [Formula: see text]-modules, every essential extension of [Formula: see text] is a direct sum of lifting modules, where [Formula: see text] denotes the injective hull. We show that every essential extension of [Formula: see text] is a direct sum of lifting modules if and only if [Formula: see text] is right Noetherian and [Formula: see text] is hollow. Assume that [Formula: see text] is an injective right [Formula: see text]-module with essential socle. We also prove that if every essential extension of [Formula: see text] is a direct sum of lifting modules, then [Formula: see text] is [Formula: see text]-injective. As a consequence of this observation, we show that [Formula: see text] is a right V-ring and every essential extension of [Formula: see text] is a direct sum of lifting modules for all simple modules [Formula: see text] if and only if [Formula: see text] is a right [Formula: see text]-V-ring.

scholarly journals Usage of Interface Management System in Adaptive Reuse of Buildings

Buildings ◽  
2019 ◽  
Vol 9 (5) ◽  
pp. 105 ◽  
Author(s):  
Ekin Eray ◽  
Benjamin Sanchez ◽  
Carl Haas
Keyword(s):  
Construction Projects ◽  
Adaptive Reuse ◽  
Project Delivery ◽  
Interface Management ◽  
Building Projects ◽  
Planning Methods ◽  
New Construction ◽  
Definition Of ◽  
Building Project

Adaptive reuse of buildings is considered a superior alternative for new construction in terms of sustainability and a disruptive practice in the current capital project delivery model for the renewal of today’s built environment. In comparison to green-field construction projects, adaptive reuse projects require distinct stages, definition of interfaces, decision gates, and planning methods in order to secure the success of the building project. Unfortunately, little research has been done regarding establishing feasible systems for the planning, assessment, and management of adaptive reuse projects, leading to underperforming building projects outcomes. Interface management (IM) can improve renovation projects outcomes by defining appropriate ways to identify, record, monitor, and track project interfaces. IM has the potential of bringing cost and time benefits during adaptive reuse projects execution. The aim of this study is to develop a reference framework for implementing IM for adaptive reuse projects. First, the inefficiencies of redevelopment projects are explained inside of a circular economy (CE) context. Second, an ontology of IM for adaptive reuse projects is defined based on the current barriers to adaptive reuse and the most common interface problems in construction projects. Third, the defined ontology is expanded through a case study by showing examples of adaptive reuse barriers on a case project, and how IM could have been part of the solution for these problems. Finally, this study concludes with the suggestions on interface management systems (IMS) implementation for future adaptive reuse projects.

Galois Theory of Essential Extensions of Modules

1972 ◽  
Vol 24 (4) ◽  
pp. 573-579 ◽  
Author(s):  
Sylvia Wiegand
Keyword(s):  
Galois Theory ◽  
Algebraic Closure ◽  
Essential Extension ◽  
Algebraic Extension ◽  
Injective Hull ◽  
Algebraic Systems ◽  
Image Position ◽  
Essential Extensions

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules(1)which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

Sign in / Sign up

Export Citation Format

Share Document