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Coneat submodules and coneat-flat modules
Abstract
A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism N -> S can be extended to a homomorphism M -> S. M is called coneat-flat if the kernel of any epimorphism Y -> M -> 0 is coneat in Y. It is proven that (1) coneat submodules of any right R-module are coclosed if and only if R is right K-ring; (2) every right R-module is coneat-flat if and only if R is right V-ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if R is right small ring. If R is commutative, then a module M is coneat-flat if and only if M+ is m-injective. Every maximal left ideal of R is finitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring R is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and fiat modules coincide.
... The usual ambient exact structure D will be that given by the class of all short exact sequences in the category. In module categories there are numerous examples of E-divisible objects: injective, pure-injective, absolutely pure (i.e., every short exact sequence starting with it is pure [20,22]), finitely injective (i.e., every short exact sequence starting with it is finitely split [2]), finitely pure-injective (i.e., every pure short exact sequence starting with it is finitely split [2]), absolutely neat (i.e., every short exact sequence starting with it is neat [13]), absolutely coneat (i.e., every short exact sequence starting with it is coneat [5]), absolutely s-pure (i.e., every short exact sequence starting with it is s-pure [10]), weak injective (i.e., every short exact sequence starting with it is closed [34]) objects etc. Each of the above E -divisible objects has a corresponding notion of E-flat object, namely projective, pureprojective, flat, finitely projective [2], finitely pure-projective [2], neat-flat [7], coneatflat [5], max-flat [31] and weak flat [35] object respectively. ...
... In module categories there are numerous examples of E-divisible objects: injective, pure-injective, absolutely pure (i.e., every short exact sequence starting with it is pure [20,22]), finitely injective (i.e., every short exact sequence starting with it is finitely split [2]), finitely pure-injective (i.e., every pure short exact sequence starting with it is finitely split [2]), absolutely neat (i.e., every short exact sequence starting with it is neat [13]), absolutely coneat (i.e., every short exact sequence starting with it is coneat [5]), absolutely s-pure (i.e., every short exact sequence starting with it is s-pure [10]), weak injective (i.e., every short exact sequence starting with it is closed [34]) objects etc. Each of the above E -divisible objects has a corresponding notion of E-flat object, namely projective, pureprojective, flat, finitely projective [2], finitely pure-projective [2], neat-flat [7], coneatflat [5], max-flat [31] and weak flat [35] object respectively. For further motivation and background the reader is referred to [14]. ...
... Let D and E be the exact structures given by short exact sequences and coneat short exact sequences in Mod(R) respectively. Then D-E-divisible and D-E-flat objects coincide with absolutely coneat [13] and coneat-flat [5] right R-modules respectively. ...
For a Quillen exact category \({\mathcal {C}}\) endowed with two exact structures \({\mathcal {D}}\) and \({\mathcal {E}}\) such that \({\mathcal {E}}\subseteq {\mathcal {D}}\), an object X of \({\mathcal {C}}\) is called \({\mathcal {E}}\)-divisible (respectively \({\mathcal {E}}\)-flat) if every short exact sequence from \({\mathcal {D}}\) starting (respectively ending) with X belongs to \({\mathcal {E}}\). We continue our study of relatively divisible and relatively flat objects in Quillen exact categories with applications to finitely accessible additive categories and module categories. We derive consequences for exact structures generated by the simple modules and the modules with zero Jacobson radical.
... Recently, Zöschinger showed in [25] that, over commutative noetherian rings, coclosed submodules in every R-module are closed if and only if R is distributive. In general coclosed submodules are coneat, [1]. Coneat submodules are coclosed if and only if R is left K-ring (i.e., every small left R-module is coatomic), [1,Theorem 2.21]. ...
... In general coclosed submodules are coneat, [1]. Coneat submodules are coclosed if and only if R is left K-ring (i.e., every small left R-module is coatomic), [1,Theorem 2.21]. ...
... Since every coclosed submodules in R-module are coneat, neat submodules in R-module are coneat by (1). Then neat and coneat submodules coincide by [7, p. 138], and hence coneat submodules in R-module are closed since R is C-ring by [8,Theorem 5]. ...
Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially different ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of c-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring R, we prove that, (1) direct product of simple R-modules is c-injective; (2) an R-module D is c-injective if and only if it is isomorphic to a direct summand of a direct product of simple R-modules and injective R-modules.
... Clearly those algebras would be Kasch if the projective dimension of the indecomposable modules in which simples embed is 0 and allowing the projective dimension to be non-zero but finite seems to be a generalization of the Kasch condition. In this paper we are considering the dual condition of the Kasch condition, namely that every simple module is a factor module of an injective module, i.e. a module of injective dimension 0. Self-injective rings and V-rings are obvious examples of dual Kasch rings and it became apparent from recent work by the first author in [6], that the dual Kasch condition can be characterized by saying that the ring R is a coneat submodule of its injective hull E(R) (see Theorem 2.3). Under additional conditions on E(R), we show that being Kasch implies being dual Kasch and conversely. ...
... Before characterizing dual Kasch rings we need some more terminology: Following [6], a submodule N of a right R-module M is said to be coneat in M if for all simple right R-modules S, any homomorphism N → S can be extended to a homomorphism M → S. By If any of these conditions hold, then any projective simple right module is injective. ...
It is well known that a ring [Formula: see text] is right Kasch if each simple right [Formula: see text]-module embeds in a projective right [Formula: see text]-module. In this paper we study the dual notion and call a ring [Formula: see text] right dual Kasch if each simple right [Formula: see text]-module is a homomorphic image of an injective right [Formula: see text]-module. We prove that [Formula: see text] is right dual Kasch if and only if every finitely generated projective right [Formula: see text]-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field [Formula: see text], a finite dimensional [Formula: see text]-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.
... Clearly those algebras would be Kasch if the projective dimension of the indecomposable modules in which simples embed is 0 and allowing the projective dimension to be non-zero but finite seems to be a generalization of the Kasch condition. In this paper we are considering the dual condition of the Kasch condition, namely that every simple module is a factor module of an injective module, i.e. a module of injective dimension 0. Self-injective rings and V-rings are obvious examples of dual Kasch rings and it became apparent from recent work by the first author in [6], that the dual Kasch condition can be characterized by saying that the ring R is a coneat submodule of its injective hull E(R) (see Theorem 2.3). Under additional conditions on E(R), we show that being Kasch implies being dual Kasch and conversely. ...
... Before characterizing dual Kasch rings we need some more terminology: Following [6], a submodule N of a right R-module M is said to be coneat in M if for all simple right R-modules S, any homomorphism N → S can be extended to a homomorphism M → S. By If any of these conditions hold, then any projective simple right module is injective. ...
It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field $k$, a finite dimensional $k$-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.
... One of the two exact structures D and E is often that given by the class of all short exact sequences in the category, but we point out that this class fails to form an exact structure unless the underlying category is not quasi-abelian (see Example 2.3). Examples of D-E-divisible objects in suitable categories include: injective, pure-injective, absolutely pure (i.e., every short exact sequence starting with it is pure [21,22]), finitely injective (i.e., every short exact sequence starting with it is finitely split [1]), finitely pure-injective (i.e., every pure short exact sequence starting with it is finitely split [1]), absolutely neat (i.e., every short exact sequence starting with it is neat [8]), absolutely coneat (i.e., every short exact sequence starting with it is coneat [4]), absolutely s-pure (i.e., every short exact sequence starting with it is s-pure [6]), weak injective (i.e., every short exact sequence starting with it is closed [35]) objects etc. Each of the above situations generates a corresponding notion of D-E-flat object, namely projective, pure-projective, flat, finitely projective [1], finitely pure-projective [1], neat-flat [5], coneat-flat [4], max-flat [33] and weak flat [36] object respectively. ...
... Examples of D-E-divisible objects in suitable categories include: injective, pure-injective, absolutely pure (i.e., every short exact sequence starting with it is pure [21,22]), finitely injective (i.e., every short exact sequence starting with it is finitely split [1]), finitely pure-injective (i.e., every pure short exact sequence starting with it is finitely split [1]), absolutely neat (i.e., every short exact sequence starting with it is neat [8]), absolutely coneat (i.e., every short exact sequence starting with it is coneat [4]), absolutely s-pure (i.e., every short exact sequence starting with it is s-pure [6]), weak injective (i.e., every short exact sequence starting with it is closed [35]) objects etc. Each of the above situations generates a corresponding notion of D-E-flat object, namely projective, pure-projective, flat, finitely projective [1], finitely pure-projective [1], neat-flat [5], coneat-flat [4], max-flat [33] and weak flat [36] object respectively. ...
We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides with the class of relatively flat objects of $\mathcal{C}$ for some relative projectively generated exact structure, while $\mathcal{B}$ coincides with the class of relatively divisible objects of $\mathcal{C}$ for some relative injectively generated exact structure. We exhibit Galois connections between relative cotorsion pairs, relative projectively generated exact structures and relative injectively generated exact structures in additive categories. We establish closure properties and characterizations in terms of approximation theory.
... Let D and E be the exact structures given by short exact sequences and coneat short exact sequences in Mod(R) respectively. Then D-E-divisible and D-E-flat objects coincide with absolutely coneat [12] and coneat-flat [5] right R-modules respectively. ...
We continue our study of relatively divisible and relatively flat objects in exact categories in the sense of Quillen with several applications to exact structures on finitely accessible additive categories and module categories. We derive consequences for exact structures generated by the simple modules and the modules with zero Jacobson radical.
... In[9], a submodule N of an R-module M is called coneat in M if for every simple R-module S, any homomorphism φ : N → S can be extended to a homomorphism θ : M → S. In[8], an R-module M is called absolutely coneat if M is a coneat submodule of any module containing it. Absolutely coneat modules are also studied in[4]. Coclosed submodules are coneat by[4, Proposition 2.1]. ...
An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingular R-modules; (3) absolutely coneat modules are strongly noncosingular if and only if R is a right max ring and injective modules are strongly noncosingular; (4) a commutative ring R is semisimple if and only if the class of injective modules coincides with the class of strongly noncosingular R-modules.
Recently, in a series of papers “simple” versions of direct-injective and direct-projective modules have been investigated. These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.
We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.
In this paper, we investigate Gorenstein FC-projective modules and Gorenstein FC-projective dimensions, and characterize rings over which every module is Gorenstein FC-projective and rings over which every submodule of a projective is Gorenstein FC-projective, respectively. Under an almost excellent extension of rings, we obtain several invariant properties of Gorenstein FC-projective modules and Gorenstein FC-projective dimensions.
An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author's attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.
Let R be a ring with an identity element. We prove that R is right Kasch if and only if injective hull of every simple right R-modules is neat-flat if and only if every absolutely pure right R-module is neat-flat. A commutative ring R is hereditary and noetherian if and only if every absolutely s-pure R-module is injective and R is nonsingular. If every simple right R-module is finitely presented, then (1) R R is absolutely s-pure if and only if R is right Kasch and (2) R is a right -CS ring if and only if every pure injective neat-flat right R-module is projective if and only if every absolutely s-pure left R-module is injective and R is right perfect. We also study enveloping and covering properties of absolutely s-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized.
We prove that neat and coneat submodules of a module coincide when \$R\$ is a commutative ring such that every maximal ideal is principal, extending a recent result by Fuchs. We characterise absolutely neat (coneat) modules and study their closure properties. We show that a module is absolutely neat if and only if it is injective with respect to the Dickson torsion theory.
Fieldhouse observed that any finitely presented left R -module P is projective with respect to pure exact sequences, i.e.
can always be completed to a commutative diagram when the sequence is pure exact. A left R -module A is absolutely pure if it is a pure submodule of every module which contains it.
Let R be a commutative, noetherian ring, and let M be an R-module. M is called weakly flat if the kernel of any epimorphism Y ↠ M is closed in Y. Equivalently, a consequence of M/U being singular is that U is essential in M, or that Ann R(𝔭)·M is essential in Ann M(𝔭) for all 𝔭 ∈AssR(M). For a series of important module classes we give an explicit description of their weakly flat objects, and in the local case we investigate the connection between the weak flatness of M and the weak injectivity of the Matlis dual M°. Finally, we characterize those rings R over which every weakly flat R-module is already flat.
Neat subgroups of abelian groups have been generalized to modules in essentially two different ways (corresponding to (a) and (b) in the Introduction); they are in general inequivalent, none implies the other. Here we consider relations between the two versions in the commutative case, and characterize the integral domains in which they coincide: these are the domains whose maximal ideals are invertible.
We characterize left coherent, left Noetherian, and left Artinian rings by the flatness, projectivity, and injectivity of the character module of certain left R-modules.