A generalization of injectivity

We study a class of modules which can be characterized using a duality theorem, called finitistic n-self-cotilting. Such a module Q can be characterized using dual conditions of some generalizations for star modules: every module M which has a right resolution with n terms isomorphic to finite powers of Q (i.e., M is n-finitely Q-copresented) has a right resolution with (n + 1) terms, and the functor Hom R (−, Q) preserves the exactness of all monomorphisms with their ranges finite powers of Q and cokernels n-finitely Q-copresented modules. In the general case, these modules are independent toward other kinds of modules which are characterized using some dualities (w-Π f -quasi injective modules, costar modules, f-cotilting modules). Closure properties for the classes involved in the duality are studied. At the end of the article, connections with the cotilting theory are exhibited, in the case of finitely dimensional algebras over fields.

A right R-module M is said to be finendo provided that as a left module over its endomorphism ring it is finitely generated. Let B = End MR. Then Bn → M → 0 is exact for some finite integer n. This implies (and when M is quasi-injective is implied by) the exactitude of 0 → R/annRM → Mn. This implies that a finendo module is quasi iff injective modulo annihilator (1). Similarly for any finitely generated module over a commutative ring. Any finendo quasi-injective module is injective over its biendomorphism ring (17B). A ring R is right Artinian iff every quasi-injective right module is finendo (17). This supplies the converse of a theorem of K. Fuller [69]. Moreover, a ring R is right self-injective iff every finendo faithful (injective) right R-module generates mod-R (18B). Azumaya defined a right PF ring as one such that every faithful right R-module generates mod-R. We characterize a right PF ring as a right self-injective ring such that every faithful (quasi) injective right R-module is finendo (38). Then, if every factor ring is PF, the ring must be right Artinian, hence uniserial (40). In general, a factor ring R/A of a right PF ring is right PF iff the left annihilator of A is Rz = zR for some z e R (39).

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module [Formula: see text] over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of [Formula: see text] extend to nilpotent endomorphisms of [Formula: see text]. It is also shown that a right module [Formula: see text] over a prime right Goldie ring with dim[Formula: see text] is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if [Formula: see text] is a finitely generated nilpotent-coinvariant right module with [Formula: see text] square-full, then [Formula: see text] is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.

Dual reciprocity boundary element formulation for potential problems with arbitrarily distributed sources

We find sufficient and necessary conditions for the category of Gorenstein projective modules of an artin algebra being an abelian category, and give another proof for the Auslander–Solberg correspondence which demonstrates the concrete form of the category of Gorenstein projective modules. Then we find a characterization for this category of Gorenstein projective modules. At last we give an example of this correspondence.

D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Here, employing a notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

Extension groups between atoms in abelian categories

Recently, several authors have adopted new alternative approaches in the study of some classical notions of modules. Among them, we find the notion of subprojectivity which was introduced to measure in a way the degree of projectivity of modules. The study of subprojectivity has recently been extended to the context of abelian categories, which has brought to light some interesting new aspects. For instance, in the category of complexes, it gives a new way to measure, among other things, the exactness of complexes. In this paper, we prove that the subprojectivity notion provides a new sight of null-homotopic morphisms in the category of complexes. This will be proven through two main results. Moreover, various results which emphasize the importance of subprojectivity in the category of complexes are also given. Namely, we give some applications by characterizing some classical rings and establish various examples that allow us to reflect the scope and limits of our results.

See related article, p 526–532 The SYMPLICITY investigators reported a 25- to 30-mm Hg decrease in office systolic blood pressure 6 to 12 months after renal sympathetic denervation in patients with resistant hypertension.1,2 In the SYMPLICITY studies,1,2 resistant hypertension was a systolic blood pressure of ≥160 mm Hg (150 mm Hg in patients with diabetes mellitus) on treatment with ≥3 different classes of antihypertensive drugs.1,2 This definition of resistant hypertension complies with current guidelines3 but is weak and urgently needs revision.4 Indeed, patients categorized as being apparently resistant according to this generic definition often cumulate clinical conditions that do not justify renal denervation as the preferred treatment option. These conditions include white-coat hypertension,5 poor adherence,6 suboptimal drug treatment, undiagnosed secondary hypertension,3 and isolated systolic hypertension in the elderly, a highly prevalent and difficult to treat form of hypertension driven by arterial stiffening rather than enhanced sympathetic drive.7 Neither the SYMPLICITY studies nor most nonrandomized observational reports published subsequently8 implemented a systematic strategy for differentiating between truly resistant hypertension and conditions that might benefit from an alternative approach, such as addressing the …

The aim of the article is to propose an account of the motivated nature of the sign, with special attention devoted to motivation in language. The starting point for the discussion is a cursory critique of the classic Saussurean model of arbitrariness and motivation. The account proposed in this article is in the spirit of broadly understood cognitive linguistics and Peircean semiotics. The model of motivation is comprehensive and specific. It is comprehensive in the sense that it provides a general definition of the term “motivation,” which attempts to cover all instances of its use discussed in modern semiotics. The model is specific in the sense that it proposes parameters of motivation, which allow for investigating different facets of the phenomenon. The article includes case studies which illustrate how the approach can used in actual analysis of semiotic data.

To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$ such that $\mathscr{A}$ has enough $\mathcal{F}$-projectives and enough $\mathcal{F}$-injectives and Quillen exact structures $\mathcal{E}$ with enough $\mathcal{E}$-projectives and enough $\mathcal{E}$-injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are $\mathcal{E}$-resolving and epimorphic precovering with kernels in their right $\mathcal{E}$-orthogonal class and subcategories which are $\mathcal{E}$-coresolving and monomorphic preenveloping with cokernels in their left $\mathcal{E}$-orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary $\mathcal{E}$-cotorsion pairs in the module category.

We study the transfer of D4 and dually C4 objects via functors between abelian categories. We also give applications to Grothendieck categories, comodule categories, (graded) module categories.