Projectivity relative to closed (neat) submodules

2021 ◽  
pp. 2250114
Author(s):  
Yilmaz Durğun
Keyword(s):  
Projective Modules ◽  
Finitely Generated ◽  
Closed Submodules ◽  
Pure Submodules ◽  
The Relationship

An [Formula: see text]-module [Formula: see text] is called closed (neat) projective if, for every closed (neat) submodule [Formula: see text] of every [Formula: see text]-module [Formula: see text], every homomorphism from [Formula: see text] to [Formula: see text] lifts to [Formula: see text]. In this paper, we study closed (neat) projective modules. In particular, the structure of a ring over which every finitely generated (cyclic, injective) right [Formula: see text]-module is closed (neat) projective is studied. Furthermore, the relationship among the proper classes which are induced by closed submodules, neat submodules, pure submodules and [Formula: see text]-pure submodules are investigated.

scholarly journals On finitely generated n-SG-projective modules

2010 ◽  
Vol 121 (1) ◽  
pp. 35-44
Author(s):  
Driss Bennis
Keyword(s):  
Projective Modules ◽  
Finitely Generated

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

Localization, Algebraic Loops and H-Spaces I

1979 ◽  
Vol 31 (2) ◽  
pp. 427-435 ◽  
Author(s):  
Albert O. Shar
Keyword(s):  
Nilpotent Group ◽  
Algebraic Structure ◽  
Nilpotent Groups ◽  
Topological Spaces ◽  
Finitely Generated ◽  
Cw Complex ◽  
Topological Localization ◽  
The Relationship

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.

3.—A Note on Projective Modules

1976 ◽  
Vol 75 (1) ◽  
pp. 24-31 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
Finitely Generated ◽  
Right Noetherian Rings

SynopsisFor various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

scholarly journals Coherency and Constructions for Monoids

2020 ◽  
Vol 71 (4) ◽  
pp. 1461-1488
Author(s):  
Yang Dandan ◽  
Victoria Gould ◽  
Miklós Hartmann ◽  
Nik Ruškuc ◽  
Rida-E Zenab
Keyword(s):  
Finiteness Condition ◽  
Direct Products ◽  
Finitely Generated ◽  
Rees Matrix Semigroups ◽  
The Relationship ◽  
Brandt Semigroups ◽  
Finitely Presented ◽  
Matrix Semigroups

Abstract A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

scholarly journals Recollement of homotopy categories and Cohen-Macaulay modules

2011 ◽  
Vol 8 (3) ◽  
pp. 507-542 ◽  
Author(s):  
Osamu Iyama ◽  
Kiriko Kato ◽  
Jun-ichi Miyachi
Keyword(s):  
Homotopy Category ◽  
Module Category ◽  
Gorenstein Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Stable Module Category ◽  
Quotient Category ◽  
Stable Module

AbstractWe study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.

scholarly journals FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

2012 ◽  
Vol 55 (1) ◽  
pp. 145-160 ◽  
Author(s):  
THOMAS HÜTTEMANN ◽  
DAVID QUINN
Keyword(s):  
Laurent Series ◽  
Chain Complex ◽  
Laurent Polynomial ◽  
Projective Modules ◽  
Finitely Generated ◽  
Formal Laurent Series ◽  
Chain Complexes ◽  
Laurent Polynomial Ring ◽  
Bounded Below ◽  
Free Modules

AbstractSuppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x−1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ⊗LR((x)) and C ⊗LR((x−1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology34(3) (1995), 619–632). Here R((x)) = R[[x]][x−1] and R((x−1)) = R[[x−1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

Reflections on some aspects of infinite groups

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Benjamin Fine ◽  
Anthony Gaglione ◽  
Gerhard Rosenberger ◽  
Dennis Spellman
Keyword(s):  
Finitely Generated ◽  
Infinite Groups ◽  
Limit Groups ◽  
Finitely Presented Groups ◽  
The Relationship ◽  
Finitely Presented

AbstractIn this paper we survey and reflect upon several aspects of the theory of infinite finitely generated and finitely presented groups that were originally motivated by work of Gilbert Baumslag. All but the last of the topics we have chosen are all related in one way or another to the theory of limit groups and the solution of the Tarski problems. These include the residually free and fully residually free properties and the big powers condition; Baumslag doubles and extensions of centralizers; residually-𝒳 groups and extensions of results of Benjamin Baumslag and finally the relationship between CT and CSA groups.

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