Research

Jump to: Review

15 pages, 290 KiB

Open AccessArticle

On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities

by Muhammad Asif, Ahmad N. Al-Kenani, Naveed Hussain and Muhammad Ahsan Binyamin

Mathematics 2023, 11(8), 1935; https://doi.org/10.3390/math11081935 - 20 Apr 2023

Viewed by 771

It is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras

L_{k}^{l} (V)

associated to isolated hypersurface singularities defined to be the Lie algebra of [...] Read more.

It is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras

L_{k}^{l} (V)

associated to isolated hypersurface singularities defined to be the Lie algebra of derivations of the local Artinian algebra

M_{n}^{l} (V) : = O_{l} / ⟨ F, J n ⟩

, i.e.,

L_{k}^{l} (V) = D e r (M_{n}^{l} (V))

. In this paper, we construct a new conjecture for the complete characterization of simple hypersurface singularities using the Lie algebras

L_{k}^{l} (V)

under certain condition and prove it true for

L_{k}^{l} (V)

when

k, l = 2

. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

5 pages, 241 KiB

Open AccessArticle

Formal Matrix Rings: Isomorphism Problem

by Piotr Krylov and Askar Tuganbaev

Mathematics 2023, 11(7), 1720; https://doi.org/10.3390/math11071720 - 04 Apr 2023

Viewed by 866

Abstract

We consider the isomorphism problem for formal matrix rings over a given ring. Principal factor matrices of such rings play an important role in this case. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

11 pages, 276 KiB

Open AccessArticle

Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension

by M. Palanikumar, Omaima Al-Shanqiti, Chiranjibe Jana and Madhumangal Pal

Mathematics 2023, 11(6), 1309; https://doi.org/10.3390/math11061309 - 08 Mar 2023

Cited by 1 | Viewed by 805

Abstract

In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial [...] Read more.

In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial orderings, infinite partial additions, and binary multiplications. In this paper, we introduce the notions of a one-prime partial bi-ideal, a two-prime partial bi-ideal, and a three-prime partial bi-ideal, as well as their extensions to partial rings, in addition to some characteristics of various prime partial bi-ideals. In this paper, we demonstrate that two-prime partial bi-ideal is a generalization of a one-prime partial bi-ideal, and three-prime partial bi-ideal is a generalization of a two-prime partial bi-ideal and a one-prime partial bi-ideal. A discussion of the

m_{p b 1}, (m_{p b 2}, m_{p b 3})

systems is presented. In general, the

m_{p b 2}

system is a generalization of the

m_{p b 1}

system, while the

m_{p b 3}

system is a generalization of both

m_{p b 2}

and

m_{p b 1}

systems. If

Φ

is a prime bi-ideal of ℧, then

Φ

is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if

℧ \ Φ

is a

m_{p b 1}

system (

m_{p b 2}

system,

m_{p b 3}

system) of ℧. If

Θ

is a prime bi-ideal in the complete partial ring ℧ and

Δ

is an

m_{p b 3}

system of ℧ with

Θ \cap Δ = ϕ

, then there exists a three-prime partial bi-ideal

Φ

of ℧, such that

Θ \subseteq Φ

with

Φ \cap Δ = ϕ

. These are necessary and sufficient conditions for partial bi-ideal

Θ

to be a three-prime partial bi-ideal of ℧. It is shown that partial bi-ideal

Θ

is a three-prime partial bi-ideal of ℧ if and only if

H_{Θ}

is a prime partial ideal of ℧. If

Θ

is a one-prime partial bi-ideal (two-prime partial bi-ideal) in ℧, then

H_{Θ}

is a prime partial ideal of ℧. It is guaranteed that a three-prime partial bi-ideal

Φ

with a prime bi-ideal

Θ

does not meet the

m_{p b 3}

system. In order to strengthen our results, examples are provided. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

12 pages, 283 KiB

Open AccessArticle

The Sharp Upper Estimate Conjecture for the Dimension δ_k(V) of New Derivation Lie Algebra

by Naveed Hussain, Ahmad N. Al-Kenani, Muhammad Arshad and Muhammad Asif

Mathematics 2022, 10(15), 2618; https://doi.org/10.3390/math10152618 - 27 Jul 2022

Cited by 2 | Viewed by 820

Abstract

Hussain, Yau, and Zuo introduced the Lie algebra

L_{k} (V)

from the derivation of the local algebra [...] Read more.

Hussain, Yau, and Zuo introduced the Lie algebra

L_{k} (V)

from the derivation of the local algebra

M_{k} (V) : = O_{n} / (g + J_{1} (g) + \dots + J_{k} (g))

. To find the dimension of a newly defined algebra is an important task in order to study its properties. In this regard, we compute the dimension of Lie algebra

L_{5} (V)

and justify the sharp upper estimate conjecture for fewnomial isolated singularities. We also verify the inequality conjecture:

δ_{5} (V) < δ_{4} (V)

for a general class of singularities. Our findings are novel and an addition to the study of Lie algebra. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

23 pages, 374 KiB

Open AccessFeature PaperArticle

On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications

by Sherali S. Ibraev, Larissa S. Kainbaeva and Angisin Z. Seitmuratov

Mathematics 2022, 10(10), 1680; https://doi.org/10.3390/math10101680 - 13 May 2022

Viewed by 1159

Abstract

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and [...] Read more.

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let

G

be a semisimple and simply connected algebraic group

G

over an algebraically closed field of characteristic

p > h,

where

h

is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of

G

by

G_{1}

and

g

, respectively. First, we calculate the restricted cohomology of

g

with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of

g

is equivalent to the corresponding cohomology of

G_{1},

we describe them as the cohomology of

G_{1}

in terms of the cohomology for

G_{1}

with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms

H^{n} (G_{1}, V) ≅ H^{n} (G, V)

and

H^{n} (g, V) ≅ H^{n} (G, V),

and a necessary condition for the isomorphism

H^{n} (g, V) ≅ H^{n} (G_{1}, V),

where

V

is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of

G

,

G_{1}

, and

g

with coefficients in the considered simple modules. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

26 pages, 369 KiB

Open AccessArticle

Satellites of Functors for Nonassociative Algebras with Metagroup Relations

by Sergey Victor Ludkowski

Mathematics 2022, 10(7), 1169; https://doi.org/10.3390/math10071169 - 04 Apr 2022

Cited by 1 | Viewed by 915

Abstract

The article is devoted to non-associative algebras with metagroup relations and modules over them. Their functors are studied. Satellites of functors are scrutinized. An exactness of satellite sequences and diagrams is investigated. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

3 pages, 205 KiB

Open AccessArticle

On Rings of Weak Global Dimension at Most One

by Askar Tuganbaev

Mathematics 2021, 9(21), 2643; https://doi.org/10.3390/math9212643 - 20 Oct 2021

Cited by 2 | Viewed by 1186

Abstract

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals [...] Read more.

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of R is distributive. Jensen has proved earlier that a commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring R is said to be centrally essential if either R is commutative or, for every noncentral element

x \in R

, there exist two nonzero central elements

y, z \in R

with

x y = z

. In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

23 pages, 338 KiB

Open AccessArticle

Gröbner–Shirshov Bases Theory for Trialgebras

by Juwei Huang and Yuqun Chen

Mathematics 2021, 9(11), 1207; https://doi.org/10.3390/math9111207 - 26 May 2021

Cited by 4 | Viewed by 2477

Abstract

We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary [...] Read more.

We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

Review

Jump to: Research

33 pages, 532 KiB

Open AccessReview

Nonassociative Algebras, Rings and Modules over Them

by Sergey Victor Ludkowski

Mathematics 2023, 11(7), 1714; https://doi.org/10.3390/math11071714 - 03 Apr 2023

Viewed by 1299

Abstract

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, [...] Read more.

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, nonassociative cyclic algebras, rings obtained as nonassociative cyclic extensions, nonassociative Ore extensions of hom-associative algebras and modules over them, and von Neumann finiteness for nonassociative algebras. Furthermore, there are outlined nonassociative algebras and rings and modules over them related to harmonic analysis on nonlocally compact groups, nonassociative algebras with conjugation, representations and closures of nonassociative algebras, and nonassociative algebras and modules over them with metagroup relations. Moreover, classes of Akivis, Sabinin, Malcev, Bol, generalized Cayley–Dickson, and Zinbiel-type algebras are provided. Sources also are reviewed on near to associative nonassociative algebras and modules over them. Then, there are the considered applications of nonassociative algebras and modules over them in cryptography and coding, and applications of modules over nonassociative algebras in geometry and physics. Their interactions are discussed with more classical nonassociative algebras, such as of the Lie, Jordan, Hurwitz and alternative types. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

74 pages, 751 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Centrally Essential Rings and Semirings

by Askar Tuganbaev

Mathematics 2022, 10(11), 1867; https://doi.org/10.3390/math10111867 - 30 May 2022

Viewed by 1278

Abstract

This paper is a survey of results on centrally essential rings and semirings. A ring (respectively, semiring) is said to be centrally essential if it is either commutative or satisfies the property that for any non-central element a, there exist non-zero central [...] Read more.

This paper is a survey of results on centrally essential rings and semirings. A ring (respectively, semiring) is said to be centrally essential if it is either commutative or satisfies the property that for any non-central element a, there exist non-zero central elements x and y with ax = y. The class of centrally essential rings is very large; many corresponding examples are given in the work. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

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