C-RINGS, COPRODUCTS, AND REFLECTION FUNCTORS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350071
Author(s):  
EDWARD L. GREEN ◽  
NICHOLAS A. LOEHR ◽  
ALLEN N. PELLEY
Keyword(s):  
Commutative Ring ◽  
Full Subcategory ◽  
Path Algebra ◽  
Ring Homomorphisms ◽  
Left And Right ◽  
Category Of Modules ◽  
Reflection Functors

The paper begins with a detailed study of the category of modules over two different rings using a coproduct construction. If C is a commutative ring and R and S are rings together with ring homomorphisms from C to R and C to S, then we show that the category of C-modules that are also left R-modules and right S-modules is equivalent to the category of left modules over the coproduct of R and S op in an appropriate category. Letting K denote a field, we apply this to show that the category of K-representations of a quiver that is reflection equivalent to a path algebra KΓ is equivalent to a full subcategory of left and right K-representations of KΓ.

scholarly journals Artinian bimodule with quasi-Frobenius bimodule of translations

2019 ◽  
Vol 29 (2) ◽  
pp. 103-119
Author(s):  
Aleksandr A. Nechaev ◽  
Vadim N. Tsypyschev
Keyword(s):  
Commutative Ring ◽  
Recurrent Sequence ◽  
Artinian Rings ◽  
Linear Recurrent Sequence ◽  
Left And Right ◽  
Additional Assumptions

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

scholarly journals An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

2020 ◽  
Author(s):  
David Barnes ◽  
J. P. C. Greenlees ◽  
Magdalena Kędziorek
Keyword(s):  
Elliptic Curve ◽  
Commutative Ring ◽  
Equivariant Cohomology ◽  
Algebraic Model ◽  
Homotopy Groups ◽  
Torsion Point ◽  
Ring Spectrum ◽  
Elliptic Cohomology ◽  
Commutative Algebras ◽  
Category Of Modules

Abstract Equipping a non-equivariant topological $$\text {E}_\infty $$ E ∞ -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take $$G=SO(2)$$ G = S O ( 2 ) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an $$\text {E}_\infty $$ E ∞ -ring spectrum. Moreover, the category of modules over that $$\text {E}_\infty $$ E ∞ -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

scholarly journals Near-rings of polynomials and polynomial functions

1980 ◽  
Vol 29 (1) ◽  
pp. 61-70 ◽  
Author(s):  
Günter Pilz ◽  
Yong-Sian so
Keyword(s):  
Commutative Ring ◽  
Universal Algebra ◽  
Polynomial Functions ◽  
Ring Homomorphisms ◽  
Direct Sum

AbstractIn this paper we investigate near-rings of polynomials and polynomial functions. After some results which belong to universal algebra we turn our attention to the familiar case of polynomials and polynomial functions over a commutative ring with identity. We study the relation between ring- and near-ring homomorphisms, and the behaviour of polynomial near-rings when the ring splits into a direct sum. A discussion of the structure of these polynomial near-rings (radical, semisimplicity) finishes this paper. These investigations are motivated by Clay and Doi (1973).

scholarly journals Finite-dimensional representations of Leavitt path algebras

2018 ◽  
Vol 30 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Ayten Koç ◽  
Murad Özaydın
Keyword(s):  
Full Subcategory ◽  
Morita Equivalence ◽  
Path Algebra ◽  
Reduction Algorithm ◽  
Quiver Representations ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Finite Dimensional ◽  
Leavitt Path Algebras ◽  
Finite Digraph

Abstract When Γ is a row-finite digraph, we classify all finite-dimensional modules of the Leavitt path algebra {L(\Gamma)} via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph Γ. The category of (unital) {L(\Gamma)} -modules is equivalent to a full subcategory of quiver representations of Γ. However, the category of finite-dimensional representations of {L(\Gamma)} is tame in contrast to the finite-dimensional quiver representations of Γ, which are almost always wild.

Automorphisms of a Certain Skew Polynomial Ring of Derivation Type

1990 ◽  
Vol 42 (6) ◽  
pp. 949-958
Author(s):  
Isao Kikumasa
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Prime Integer ◽  
Ring Homomorphisms ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we setA = F[ϰ]/(ϰp - α)F[ϰ].Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. [2]).In [3], R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained ([1], [2] and [5]). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer [5] established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto [2], among others, have made some progress in case of derivation type.

scholarly journals Bar Category of Modules and Homotopy Adjunction for Tensor Functors

2020 ◽  
Author(s):  
Rina Anno ◽  
Timothy Logvinenko
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Categories ◽  
Intended Application ◽  
Adjunction Theory ◽  
Left And Right ◽  
Category Of Modules

Abstract Given a differentially graded (DG)-category ${{\mathcal{A}}}$, we introduce the bar category of modules ${\overline{\textbf{{Mod}}}-{\mathcal{A}}}$. It is a DG enhancement of the derived category $D({{\mathcal{A}}})$ of ${{\mathcal{A}}}$, which is isomorphic to the category of DG ${{\mathcal{A}}}$-modules with ${A_{\infty }}$-morphisms between them. However, it is defined intrinsically in the language of DG categories and requires no complex machinery or sign conventions of ${A_{\infty }}$-categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop a homotopy adjunction theory for tensor functors between derived categories of DG categories. It allows us to show in an enhanced setting that given a functor $F$ with left and right adjoints $L$ and $R$, the functorial complex $FR \xrightarrow{F{\operatorname{act}}{R}} FRFR \xrightarrow{FR{\operatorname{tr}} - {\operatorname{tr}}{FR}} FR \xrightarrow{{\operatorname{tr}}} {\operatorname{Id}}$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of $F$. We then write down four induced functorial Postnikov systems computing this convolution.

MULTIPLIER BI- AND HOPF ALGEBRAS

2010 ◽  
Vol 09 (02) ◽  
pp. 275-303 ◽  
Author(s):  
K. JANSSEN ◽  
J. VERCRUYSSE
Keyword(s):  
Hopf Algebra ◽  
Commutative Ring ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Left And Right ◽  
Projective Algebra

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

scholarly journals On Hyper–Regularity and Unimodularity of Ore Polynomial Matrices

2018 ◽  
Vol 28 (3) ◽  
pp. 583-594 ◽  
Author(s):  
Klemens Fritzsche ◽  
Klaus Rӧbenack
Keyword(s):  
Commutative Ring ◽  
Meromorphic Functions ◽  
Polynomial Matrices ◽  
Left And Right

Abstract We investigate Ore polynomial matrices, i. e., matrices with polynomial entries in d/dt whose coefficients are meromorphic functions in t and as such constitute a non-commutative ring. In particular, we study the properties of hyper-regularity and unimodularity of such matrices and derive conditions which make it possible to efficiently check for these characteristics. In addition, this approach enables computation of hyper-regular left and right and unimodular inverses.

scholarly journals On generalizations of injective modules

2016 ◽  
Vol 99 (113) ◽  
pp. 249-255 ◽  
Author(s):  
Burcu Türkmen
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Serial Ring ◽  
Injective Modules ◽  
Left And Right ◽  
Proper Generalization

As a proper generalization of injective modules in term of supplements, we say that a module M has the property (SE) (respectively, the property (SSE)) if, whenever M ( N, M has a supplement that is a direct summand of N (respectively, a strong supplement in N). We show that a ring R is a left and right artinian serial ring with Rad(R)2 = 0 if and only if every left R-module has the property (SSE). We prove that a commutative ring R is an artinian serial ring if and only if every left R-module has the property (SE).

The three-dimensional structure of frozen-hydrated left- and right-handed forms of bacterial flagellar filaments from an identical serotype

1986 ◽  
Vol 44 ◽  
pp. 298-299
Author(s):  
S. Trachtenberg ◽  
D. J. DeRosier
Keyword(s):  
Ionic Strength ◽  
Basal Body ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Protein Species ◽  
Liquid Environment ◽  
Bacterial Flagellum ◽  
Left And Right ◽  
Rotary Motor ◽  
Helical Parameters

The bacterial cell is propelled through the liquid environment by means of one or more rotating flagella. The bacterial flagellum is composed of a basal body (rotary motor), hook (universal coupler), and filament (propellor). The filament is a rigid helical assembly of only one protein species — flagellin. The filament can adopt different morphologies and change, reversibly, its helical parameters (pitch and hand) as a function of mechanical stress and chemical changes (pH, ionic strength) in the environment.

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