scholarly journals Semi -T- Small Submodules

2021 ◽  
pp. 1956-1960
Author(s):  
Alaa A. Elewi
Keyword(s):  
Left Module

Let  be a ring with identity and  be a submodule of a left - module . A submodule  of  is called - small in  denoted by , in case for any submodule  of ,  implies .  Submodule  of  is called semi -T- small in , denoted by , provided for submodule  of ,  implies that . We studied this concept which is a generalization of the small submodules and obtained some related results

scholarly journals P-Semi Hollow-Lifting Modules

2021 ◽  
Vol 17 (3) ◽  
pp. 37-48
Author(s):  
Mukdad Qaess Hussain ◽  
Darya Jabar Abdul Kareem
Keyword(s):  
Left Module ◽  
Basic Properties ◽  
Comprehensive Study

Let R be a ring with identity and Q be a unitary left Module over R. In this paper, we introduced the concept of p-semi hollow-lifting Module as generalization of semi hollow-lifting Module. Also, give a comprehensive study of basic properties of p-semi hollow-lifting Modules and some related concepts.

A Generalization of the Concept of a Ring of Quotients

1971 ◽  
Vol 14 (4) ◽  
pp. 517-529 ◽  
Author(s):  
John K. Luedeman
Keyword(s):  
Left Ideal ◽  
Quotient Ring ◽  
Original Method ◽  
Duke Math ◽  
Injective Hull ◽  
Direct Products ◽  
Left Module ◽  
Matrix Rings ◽  
Ring Of Quotients ◽  
Quotient Rings

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

scholarly journals Cotorsion modules and relative pure-injectivity

2006 ◽  
Vol 81 (2) ◽  
pp. 225-244 ◽  
Author(s):  
Lixin Mao ◽  
Nanqing Ding
Keyword(s):  
Left Module ◽  
Injective Modules

AbstractLet R be a ring. A right R-module C is called a cotorsion module if Ext1R (F, C) = 0 for any flat right R-module F. In this paper, we first characterize those rings satisfying the condition that every cotorsion right (left) module is injective with respect to a certain class of right (left) ideals. Then we study relative pure-injective modules and their relations with cotorsion modules.

scholarly journals A LERAY SPECTRAL SEQUENCE FOR NONCOMMUTATIVE DIFFERENTIAL FIBRATIONS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350015 ◽  
Author(s):  
EDWIN BEGGS ◽  
IBTISAM MASMALI
Keyword(s):  
Spectral Sequence ◽  
Sheaf Cohomology ◽  
Left Module ◽  
Graded Algebras ◽  
Serre Spectral Sequence ◽  
Differential Graded Algebras ◽  
Zero Curvature ◽  
Total Differential ◽  
Leray Spectral Sequence ◽  
Special Case

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals Completely Faithful Modules and Self-Injective Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 697-708 ◽  
Author(s):  
Goro Azumaya
Keyword(s):  
Endomorphism Ring ◽  
Finitely Generated ◽  
Left Module

A left module over a ring Λ is called completely faithful if Λ is a sum of those left ideals which are homomorphic images of M. The notion was first introduced by Morita [9], and he proved, among others, the following theorem which plays a basic role in his theory of category-isomorphisms: if a Λ-module M is completely faithful, then M is finitely generated and projective with respect to the endomorphism ring Γ of M and Λ coincides with the endomorphism ring of Λ-module M.

THE QUANTUM QUARTER PLANE AND THE REAL QUANTUM PLANE

2002 ◽  
Vol 13 (03) ◽  
pp. 279-321 ◽  
Author(s):  
KONRAD SCHMÜDGEN
Keyword(s):  
Fourier Transforms ◽  
Pseudodifferential Operators ◽  
Quantum Plane ◽  
Left Module ◽  
The Real ◽  
Weyl Calculus ◽  
Positive Linear Functionals ◽  
Quarter Plane ◽  
The Fourier Transform ◽  
Module Algebras

Suppose q≠±1 is a complex number of modulus one. Let [Formula: see text] be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using Hilbert space representations of [Formula: see text] and the Weyl calculus of pseudodifferential operators we construct *-algebras of "functions" on the quantum quarter plane [Formula: see text] and on the real quantum plane [Formula: see text] which are left module *-algebras for the Hopf *-algebra [Formula: see text]. We define covariant positive linear functionals hk, k∈ℤ2, and study the actions of the *-algebras [Formula: see text] and [Formula: see text] on the associated Hilbert spaces. Quantum analogs of the partial Fourier transforms and the Fourier transform are found. A differential calculus on the "function" *-algebras is also developed and investigated.

scholarly journals The "hit" problem of five variables in the generic degree and its application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Efficient Method ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Left Module ◽  
Prime Field ◽  
Hit Problem

Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree $1$. We are interested in the {\it Peterson "hit" problem} of finding a minimal set of generators for $P_s$ as a graded left module over the mod-2 Steenrod algebra, $\mathscr {A}$. For $s\geqslant 5,$ it is still open.In this paper, we study the hit problem of five variables in a generic degree. By using this result, we survey Singer's conjecture for the fifth algebraic transfer in the respective degrees. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer

scholarly journals Semi-T-Hollow Modules and Semi-T-Lifting Modules

2021 ◽  
pp. 2357-2361
Author(s):  
Alaa A. Elewi
Keyword(s):  
Direct Summand ◽  
Associative Ring ◽  
Left Module ◽  
Small Submodule

Let be an associative ring with identity and let be a unitary left -module. Let  be a non-zero submodule of .We say that  is a semi- - hollow module if for every submodule  of  such that  is a semi- - small submodule ( ). In addition, we say that  is a semi- - lifting module if for every submodule  of , there exists a direct summand  of  and  such that   The main purpose of this work was to develop the properties of these classes of module.  

scholarly journals Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 28
Author(s):  
Metod Saniga ◽  
Edyta Bartnicka
Keyword(s):  
Quantum Information ◽  
Geometric Structure ◽  
Associative Ring ◽  
Generalized Quadrangle ◽  
Two Dimensional ◽  
Left Module ◽  
Free Left ◽  
Cyclic Submodules ◽  
Free Cyclic Submodules

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

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