Direct Summand Research Articles

Flat modules and coherent endomorphism rings relative to some matrices

<abstract><p>Let $ N $ be a left $ R $-module with the endomorphism ring $ S = \text{End}(_{R}N) $. Given two cardinal numbers $ \alpha $ and $ \beta $ and a matrix $ A\in S^{\beta\times\alpha} $, $ N $ is called flat relative to $ A $ in case, for each $ x\in l_{N^{(\beta)}}(A) = \{u\in N^{(\beta)} \mid uA = 0\} $, there are a positive integer $ k $, $ y\in N^{k} $ and a $ k\times \beta $ row-finite matrix $ C $ over $ S $ such that $ CA = 0 $ and $ x = yC $. It is shown that $ N_{S} $ is flat relative to a matrix $ A $ if and only if $ l_{N^{(\beta)}}(A) $ is generated by $ N $. $ S $ is called left coherent relative to $ A $ if Ker$ (_{S}S^{(\beta)}\to _{S}S^{(\beta)}A) $ is finitely generated. It is shown that $ S $ is left coherent relative to $ A $ if and only if Hom$ _{R}(N, l_{N^{n}}(A)) $ is a finitely generated left $ S $-module if and only if $ l_{N^{n}}(A) $ has an add$ (N) $-precover (add$ (N) $ denotes the category of all direct summands of finite direct sums of copies of $ _{R}N $). Regarding applications, new necessary and sufficient conditions for epic (monic, having the unique mapping property) add$ (N) $-precovers of $ l_{N^{(\beta)}}(A) $ are investigated. Also, some new characterizations of left $ n $-semihereditary rings and von Neumann regular rings are given.</p></abstract>

MODULES WITH THE PROPERTY Radg

In this article, we introduce modules with the property Radg and provide various properties of this module. Firstly, we prove that every semisimple module has the property Radg. We also indicate that the class of modules with the property Radg is closed under finite direct sum and direct summands. Moreover, we define the concepts of g –(radical) cover. We show that factor modules of a module with the property Radg have the property Radg under special condition. Additionally, we characterize semisimple commutative rings via modules with the property Radg.

CS-Rickart and dual CS-Rickart objects in abelian categories

We introduce relative CS-Rickart objects in abelian categories, as common generalizations of relative Rickart objects and extending objects. We study direct summands and (co)products of relative CS-Rickart objects as well as classes all of whose objects are self-CS-Rickart. Corresponding results for dual relative CS-Rickart objects may be automatically obtained by the duality principle. Applications are given to Grothendieck categories and, in particular, to module and comodule categories.

Contravariant Finiteness and Iterated Strong Tilting

Let P<∞(Λ-mod)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty } ({\\Lambda }\ ext {-mod})$\\end{document} be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra Λ. We develop a widely applicable criterion that reduces the test for contravariant finiteness of P<∞(Λ-mod)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty } ({\\Lambda }\ ext {-mod})$\\end{document} in Λ-mod to corner algebras eΛe for suitable idempotents e ∈Λ. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of P<∞(Λ-mod)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty } ({\\Lambda }\ ext {-mod})$\\end{document}. The consequences pursued here hinge on the fact that this finiteness condition is known to be equivalent to the existence of a strong tilting object in Λ-mod. We moreover characterize the situation in which the process of strongly tilting Λ-mod allows for unlimited iteration: This occurs precisely when, in the category mod-Λ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ ext {mod-}\\widetilde {\\Lambda }$\\end{document} of right modules over the strongly tilted algebra Λ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\widetilde {\\Lambda }$\\end{document}, the subcategory of modules of finite projective dimension is in turn contravariantly finite; the latter condition can, once again, be tested on suitable corners eΛe of the original algebra Λ. In the (frequently occurring) positive case, the sequence of consecutive strong tilts, Λ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\widetilde {\\Lambda }$\\end{document}, Λ~~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\widetilde {\\widetilde {\\Lambda }}$\\end{document}, Λ~~~,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\widetilde {\\widetilde {\\widetilde {\\Lambda }}}, \\dots $\\end{document}, is shown to be periodic with period 2 (up to Morita equivalence); moreover, any two adjacent categories in the sequence P<∞(mod-Λ~)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty } (\ ext {mod-}\\widetilde {\\Lambda })$\\end{document}, P<∞(Λ~~-mod)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty }(\\widetilde {\\widetilde {\\Lambda }}\ ext {-mod})$\\end{document}, P<∞(mod-Λ~~~),…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty }(\ ext {mod-}\\widetilde {\\widetilde {\\widetilde {\\Lambda }}}), \\dots $\\end{document}, alternating between right and left modules, are dual via contravariant Hom-functors induced by tilting bimodules which are strong on both sides. Our methods rely on comparisons of right P<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {P}^{<\\infty }$\\end{document}-approximations in the categories Λ-mod, eΛe-mod and the Giraud subcategory of Λ-mod determined by e; these interactions hold interest in their own right. In particular, they underlie our analysis of the indecomposable direct summands of strong tilting modules.

Reflective and coreflective subcategories

Given any additive category C with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory B is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in B has a pseudocokernel in C that belongs to B. We apply this result and its dual to, among others, preabelian and pretriangulated categories. As a consequence, we show that a subcategory of a preabelian category is coreflective if, and only it, it is precovering and closed under taking cokernels. On the other hand, if C is pretriangulated with split idempotents, then a subcategory B is coreflective and invariant under the suspension functor if, and only if, it is precovering and closed under taking direct summands and cones. These are extensions of well-known results for AB3 abelian and triangulated categories, respectively.By-side applications of these results allow us: a) To characterize the coreflective subcategories of a given AB3 abelian category which have a set of generators and are themselves abelian, abelian exact or module categories; b) to extend to module categories over arbitrary small preadditive categories a result of Gabriel and De la Peña stating that all fully exact subcategories are bireflective; c) to show that, in any Grothendieck category, the direct limit closure of its subcategory of finitely presented objects is a coreflective subcategory.

Modules whose z-closed submodules are essentially embedded in summands

In this paper, we work out modules with the property that every [Formula: see text]-closed submodules are essentially embedded in direct summands. This class of modules properly contains the class of CS-modules as well as some of their generalizations. It is shown that the class of modules with former property is closed under direct sums under a certain condition. Taking into account this case, we obtain several results on the inheritance of the latter closure property. Moreover, we consider the behavior of this new property for extensions of modules.

Non-trivial smooth families of K3 surfaces

Let X be a complex K3 surface, $$\textrm{Diff}(X)$$ the group of diffeomorphisms of X and $$\textrm{Diff}_0(X)$$ the identity component. We prove that the fundamental group of $$\textrm{Diff}_0(X)$$ contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg–Witten invariants. The moduli space of Einstein metrics on X is used as a key ingredient in the proof.

Amalgamation rings and the fully invariant extending property

A module M is said to be FI-extending if every fully invariant submodule of M is essential in a direct summand of M. A ring R is right FI-extending if every ideal of R is right essential in an idempotent generated right ideal of R. A ring R is called quasi-Baer if the right annihilator of every ideal is generated as a right ideal, by an idempotent. In this paper we characterize the amalgamation ring , of the rings A, B along an ideal K of B with respect to a ring homomorphism , which is either right FI-extending or quasi-Baer.

On modules whose coclosed submodules are isomorphic to direct summands

Inspired by the notion of virtually extending module, in this work we have proposed and investigated the behaviours of virtually lifting module as the dual of virtually extending module. Several properties of this structure have been discussed and got that virtually lifting module is a \(D_{12}\) module. It is observed that for a V ring R, a module M is virtually lifting iff it is lifting. Moreover, we have defined virtually \(D_{2}\) module as a proper extension of \(D_{2}\) module, and seen that virtually \(D_{2}\) modules is inherited by direct summands. Finally, we have investigated principally h-lifting module and proved that finite direct sum of principally h-lifting modules are principally h-lifting if each components are principally relatively projective.

Pure-direct-projective modules

In this paper, we introduce and study the pure-direct-projective modules, that is the modules [Formula: see text] every pure submodule [Formula: see text] of which with [Formula: see text] isomorphic to a direct summand of [Formula: see text] is a direct summand of [Formula: see text]. We characterize rings over which every right [Formula: see text]-module is pure-direct-projective. We examine for which rings or under what conditions pure-direct-projective right [Formula: see text]-modules are direct-projective, projective, quasi-projective, pure-projective, flat or injective. We prove that over a Noetherian ring every injective module is pure-direct-projective and a right hereditary ring [Formula: see text] is right Noetherian if and only if every injective right [Formula: see text]-module is pure-direct-projective. We obtain some properties of pure-direct-projective right [Formula: see text]-modules which have [Formula: see text] and [Formula: see text].

Bivariant Class of Degree One

Let f:Xrightarrow Y be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class theta in H^0(X{mathop {rightarrow }limits ^{f}}Y)cong Hom_{D^{b}_{c}(Y)}(Rf_*{mathbb {A}}_X, {mathbb {A}}_Y) (here {mathbb {A}} is a Noetherian commutative ring with identity, and {mathbb {A}}_X and {mathbb {A}}_Y denote the constant sheaves). Let theta _0:H^0(X)rightarrow H^0(Y) be the induced Gysin morphism. We say that theta has degree one if theta _0(1_X)= 1_Yin H^0(Y). This is equivalent to say that theta is a section of the pull-back f^*: {mathbb {A}}_Yrightarrow Rf_*{mathbb {A}}_X, i.e. theta circ f^*={text {id}}_{{mathbb {A}}_Y}, and it is also equivalent to say that {mathbb {A}}_Y is a direct summand of Rf_*{mathbb {A}}_X. We investigate the consequences of the existence of a bivariant class of degree one. We prove explicit formulas relating the (co)homology of X and Y, which extend the classic formulas of the blowing-up. These formulas are compatible with the duality morphism. Using which, we prove that the existence of a bivariant class theta of degree one for a resolution of singularities, is equivalent to require that Y is an {mathbb {A}}-homology manifold. In this case theta is unique, and the Betti numbers of the singular locus {text {Sing}}(Y) of Y are related with the ones of f^{-1}({text {Sing}}(Y)).

Direct complements almost unique

Direct complements in a right [Formula: see text]-module [Formula: see text] are said to be almost unique if, whenever [Formula: see text], then [Formula: see text] (also [Formula: see text], by symmetry). We will show that this new class of modules lies strictly between the dual-square-free and the summand-dual-square-free modules. While it is an open question whether right quasi-duo (equivalently, right dual-square-free) rings are left-right symmetric, we will prove that both notions “summand-dual-square-free” and “direct complements almost unique” are left-right symmetric. Furthermore, we will show that direct complements in a module [Formula: see text] are almost unique iff idempotents in [Formula: see text] are central modulo [Formula: see text], where [Formula: see text]. As an immediate consequence, if [Formula: see text] is an epi-projective module, then direct complements in [Formula: see text] are almost unique iff [Formula: see text] is strongly perspective (i.e. if [Formula: see text] and [Formula: see text] are isomorphic direct summands of [Formula: see text] and [Formula: see text], then [Formula: see text]. In particular, direct complements in a ring [Formula: see text] are almost unique iff [Formula: see text] is strongly perspective. Moreover, if [Formula: see text] is a module with the finite exchange whose direct complements are almost unique, then [Formula: see text] is clean, strongly perspective, and satisfy the full exchange.

Principally ⨁-G-〖Rad〗_g-supplemented modules

In this paper, a new concept has been presented that is a stronger than a previous concept called -G--supplemented. The basic definition is in an ring where an -module is said to be principally -G--supplemented (shortly, -PG--supplemented) if any cyclic submodule of with , there exists a direct summand of such that and . A set of properties and relations between previous modules and the given module has been dealt with simple examples illustrating those relations.

The Continuous-Time Singular LQR Problem and the Riddle of Nonautonomous Hamiltonian Systems: A Behavioral Solution

In this article, we deal with the continuous-time singular linear quadratic regulator (LQR) problems, which give rise to nonautonomous Hamiltonian systems. This case arises when the system’s transfer function matrix is not left-invertible. A special case of this problem can be solved using the constrained generalized continuous algebraic Riccati equation (CGCARE), when a certain condition on the input-cardinality of the Hamiltonian is satisfied. However, this condition is only a special case among many other possible cases. On the other hand, singular LQR problems with autonomous Hamiltonian systems have been well studied in the literature. In this article, we apply behavioral theoretic techniques to show that the general case of the singular LQR problem with nonautonomous Hamiltonian can be solved by a direct sum decomposition of the plant behavior, where one of the direct summands can be solved via CGCARE, while the other gives rise to an autonomous Hamiltonian system.

Isomorphisms of spectral lattices

The paper deals with spectral order isomorphisms between certain spectral sublattices of direct sums of $$AW$$ *-factors. We prove that these maps consist of spectral order isomorphisms between spectral sublattices of individual direct summands. Consequently, we obtain a complete description of spectral order isomorphisms in the case of atomic $$AW$$ *-algebras. This includes the setting of matrix algebras. Moreover, we also exhibit the general form of spectral order orthoisomorphisms between various spectral sublattices of direct sums of $$AW$$ *-factors.

Tilting Preenvelopes and Cotilting Precovers in General Abelian Categories

We consider an arbitrary Abelian category \(\mathcal {A}\) and a subcategory \(\mathcal {T}\) closed under extensions and direct summands, and characterize those \(\mathcal {T}\) that are (semi-)special preenveloping in \(\mathcal {A}\); as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories \(\mathcal {T}\) of \(\mathcal {A}\) closed under extensions and direct summands are precisely those for which \(({}^{\perp _{1}}\mathcal {T},\mathcal {T})\) is a right complete cotorsion pair, where \({}^{\perp _{1}}\mathcal {T} := \text {Ker} (\text {Ext}_{\mathcal {A}}^{1}(-,\mathcal {T}))\). Particular cases appear when \(\mathcal {T}=V^{\perp _{1}}:=\text {Ker} (\text {Ext}_{\mathcal {A}}^{1}(V,-))\), for an Ext1-universal object V such that \(\text {Ext}_{\mathcal {A}}^{1}(V,-)\) vanishes on all (existing) coproducts of copies of V. For many choices of \(\mathcal {A}\), we show that these latter examples exhaust all the possibilities. We then show that, when \(\mathcal {A}\) has an epi-generator, the (semi-)special preenveloping torsion classes \(\mathcal {T}\) given by (quasi-)tilting objects are exactly those for which any object \(T\in \mathcal {T}\) is the epimorphic image of some object in \({}^{\perp _{1}}\mathcal {T}\) (and the subcategory \({\mathscr{B}}:=\text {Sub}(\mathcal {T})\) of subobjects of objects in \(\mathcal {T}\) is reflective) and they are, in turn, the right constituents of complete cotorsion pairs in \(\mathcal {A}\) (resp., \({\mathscr{B}}\)). In a final section, we apply the results when \(\mathcal {A}=\text {mod-}R\) is the category of finitely presented modules over a right coherent ring R, something that gives new results and raises new questions even at the level of classical tilting theory in categories of modules.

Abelian groups in which every neat subgroup is a direct summand

Abelian groups in which every neat subgroup is a direct summand

On groups every subgroup of which is a direct summand

On groups every subgroup of which is a direct summand

Abelian groups in which every serving subgroup is a direct summand

Abelian groups in which every serving subgroup is a direct summand

On pure subgroups and direct summands of abelian groups

On pure subgroups and direct summands of abelian groups