Algebraic Identities Research Articles

IDENTITIES WITH MULTIPLICATIVE GENERALIZED (α,α)-DERIVATIONS OF SEMIPRIME RINGS

Let R be a semiprime ring and α be an automorphism of R. A mapping F : R → R (not necessarily additive) is called multiplicative generalized (α,α)-derivation if there exists a unique (α,α)-derivation d of R such that F(xy) = F(x)α(y) + α(x)d(y) for all x,y ∈ R. In the present paper, we intend to study several algebraic identities involving multiplicative generalized (α,α)-derivations on appropriate subsets of semiprime rings and collect the information about the commutative structure of these rings.

Associated Prime Ideals in Commutative Algebra

This paper aims to explain how an abstract mathematical object of higher mathematics is learnt and solved the given problem from the pedagogical perspectives resting the explanation based upon the solution of higher mathematical problem in abstract algebra.The purpose is not only to disseminate the solution of the problem but to unfold the learning realities in course of pursuing the pure discovery learning path. The problem of the study was connected to the understandings of associated prime ideals in commutative algebra. Factual materials including books, lecture notes and research articles were used to explore the concepts on associated prime ideals. The concepts learned were summarized, conceptualized, and then analyzed together with established facts of algebraic concepts and developed examples to interpret the abstract concepts: the associated prime ideals. The associated prime ideal was explored and understood within the conceptions of primary decomposition, Noetherian rings, and modules, and then characterized some fundamental notions of it on an A-module M over the Noetherian ring A. The results reveal that repeated reading, writing, rigorous thinking and linking, and engaging on abstraction are the basic characteristics of content researcher in developing and constructing new insights, ideas and knowledge in higher mathematics from pedagogic perspective

Quotient rings satisfying some identities

This paper investigates the commutativity of the quotient ring \(\mathcal{R}/P\), where \(\mathcal{R}\) is an associative ring with a prime ideal \(P\), and the possibility of forms of derivations satisfying certain algebraic identities on \(\mathcal{R}\). We provide some results for strong commutativity-preserving derivations of prime rings.

Identities with generalized derivations on Lie ideals and Banach algebras

Abstract Let 𝑅 be a prime ring and 𝐿 a non-central Lie ideal of 𝑅. In this paper, we aim to classify the generalized derivations of 𝑅 satisfying some algebraic identities with power values on 𝐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra.

Cycle algebras and polytopes of matroids

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

Guided e-Assessment of Math Proofs with the Halomda Platform

The mode of assessment is one of the most important factors influencing learning. E-assessment usually includes only checking the final answer, thus limiting teacher's ability to check the complete solution, and it does not allow inclusion of math proofs problems that constitute an important part of math content. The e-assessment module of Halomda elearning platform allows assessment of proof problems from different areas of school and high Math, like algebraic identities and inequalities, geometry proofs, trigonometrical identities, analytic geometry, vector algebra, etc. Although eassessment of a proof of a mathematical statement is generally a difficult task, in this paper we show through some working examples how guided e-assessment can be implemented using the Halomda platform.

Classification of additive mappings on certain rings and algebras

The objective of this research is to prove that an additive mapping Δ:A→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta :{\\mathcal {A}}\\rightarrow {\\mathcal {A}}$$\\end{document} will be a generalized derivation associated with a derivation ∂:A→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial :{\\mathcal {A}}\\rightarrow {\\mathcal {A}}$$\\end{document} if it satisfies the following identity Δ(rm+n+p)=Δ(rm)rn+p+rm∂(rn)rp+rm+n∂(rp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (r^{m+n+p})=\\Delta (r^m)r^{n+p}+r^m\\partial (r^{n})r^p+r^{m+n}\\partial (r^p)$$\\end{document} for all r∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\in {\\mathcal {A}}$$\\end{document}, where m,n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m, n\\ge 1$$\\end{document} and p≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 0$$\\end{document} are fixed integers and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} is a semiprime ring. Another analogous has been done where an additive mapping behaves like a generalized left derivation associated with a left derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} satisfying certain algebraic identity. The proofs of these advancements are derived employing algebraic concepts. These theorems have been validated by offering an example that shows they are not insignificant. Furthermore, we provide an application in the framework of Banach algebra.

A general class of linear unconditionally energy stable schemes for the gradient flows, II

This paper continues to study linear and unconditionally modified-energy stable (abbreviated as SAV-GL) schemes for the gradient flows. The schemes are built on the SAV technique and the general linear time discretizations (GLTD) as well as the extrapolation for the nonlinear term. Different from [44], the GLTDs with three parameters discussed here are not necessarily algebraically stable. Some algebraic identities are derived by using the method of undetermined coefficients and further used to establish the modified-energy inequalities for the unconditional modified-energy stability of the semi-discrete-in-time SAV-GL schemes. It is worth emphasizing that those algebraic identities or energy inequalities are not necessarily unique for some choices of three parameters in the GLTDs. Numerical experiments on the Allen-Cahn, the Cahn-Hilliard and the phase field crystal models with the periodic boundary conditions are conducted to validate the unconditional modified-energy stability of the SAV-GL schemes, where the Fourier pseudo-spectral method is employed in space with the zero-padding to eliminate the aliasing error and the time stepsizes for ensuring the original-energy monotonous decay are estimated by using the stability regions of our SAV-GL schemes for the test equation. The resulting time stepsize constraints for the SAV-GL schemes are almost consistent with the numerical results on the above gradient flow models.

Commutativity with algebraic identities involving hermetian and skew-hermetian elements

This paper aims to investigate the commutativity of prime rings with involution * of the second kind in which endomorphisms satisfy some algebraic identities involving hermitian and skew-hermitian elements. We will also provide some classifications of these endomorphisms. Finally, we give some examples to prove that the imposed hypotheses are necessary.

Algebraic identities between families of (elliptic) modular graphs

Consider an algebraic identity between elliptic modular graphs where several vertices are at fixed locations (and hence unintegrated) while the others are integrated over the toroidal worldsheet. At any unintegrated vertex, we can glue an arbitrary expression involving elliptic modular graphs which has the same unintegrated vertex. Integrating over that vertex, we obtain new algebraic identities between elliptic modular graphs. Hence this elementary process of convoluting the original “seed” identity with other graphs yields infinite number of new identities. We consider various seed identities in which two of the vertices are unintegrated. Convoluting them with families of elliptic modular graphs, we obtain new identities. Each identity is parametrized by an arbitrary number of links in the graphs as well as the positions of unintegrated vertices. On identifying the unintegrated vertices, this leads to an algebraic identity involving modular graphs where all the vertices are integrated over the worldsheet.

Generalized derivations acting on Lie ideals in prime rings and Banach algebras

Let $R$ be a prime ring and $L$ a non-central Lie ideal of $R.$ The purpose of this paper is to describe generalized derivations of $R$ satisfying some algebraic identities locally on $L.$ More precisely, we consider two generalized derivations $F_1$ and $F_2$ of a prime ring $R$ satisfying one of the following identities:1. $F_1(x)\circ y +x \circ F_2(y) =0,$2. $[F_1(x),y] + F_2([x,y]) =0,$for all $x,y$ in a non-central Lie ideal $L$ of $R.$ Furthermore, as an application, we study continuous generalized derivations satisfying similar algebraic identities with power values on nonvoidopen subsets of a prime Banach algebra $A$. Our topological approach is based on Baire'scategory theorem and some properties from functional analysis.

On generalized homoderivations of prime rings

Let $\mathscr{A}$ be a ring with its center $\mathscr{Z}(\mathscr{A}).$ An additive mapping $\xi\colon \mathscr{A}\to \mathscr{A}$ is called a homoderivation on $\mathscr{A}$ if
 $\forall\ a,b\in \mathscr{A}\colon\quad \xi(ab)=\xi(a)\xi(b)+\xi(a)b+a\xi(b).$
 An additive map $\psi\colon \mathscr{A}\to \mathscr{A}$ is called a generalized homoderivation with associated homoderivation $\xi$ on $\mathscr{A}$ if
 $\forall\ a,b\in \mathscr{A}\colon\quad\psi(ab)=\psi(a)\psi(b)+\psi(a)b+a\xi(b).$
 This study examines whether a prime ring $\mathscr{A}$ with a generalized homoderivation $\psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities:
 $\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),$
 $\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ab\in \mathscr{Z}(\mathscr{A}),$
 $\psi(ab)+ba\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ba\in \mathscr{Z}(\mathscr{A})\quad (\forall\ a, b\in \mathscr{A}).$
 Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

IDEALS OF FUNCTION SPACE IN THE LIGHT OF AN EXPONENTIAL ALGEBRA

Exponential algebra is a new algebraic structure consisting of a semigroup structure, a scalar multiplication, an internal multiplication and a partial order [introduced in [4]]. This structure is based on the structure `exponential vector space' which is thoroughly developed by Priti Sharma et. al. in [11] [This structure was actually proposed by S. Ganguly et. al. in [1] with the name `quasi-vector space'] Exponential algebra can be considered as an algebraic ordered extension of the concept of algebra. In the present paper we have shown that the function space $ C^+(\mathbf X) $ of all non-negative continuous functions on a topological space $\mathbf X$ is a topological exponential algebra under the compact open topology. Also we have discussed the ideals and maximal ideals of $ C^+(\mathbf X) $. We find an ideal of $ C^+(\mathbf X)$ which is not a maximal ideal in general; actually maximality of that ideal depends on the topology of $\mathbf{X}$. The concept of ideals of exponential algebra was introduced by us in [4].

On generalized derivations involving prime ideals with involution

UDC 512.5 We study the structure of the quotient A / P , where A is any ring with involution * and P is a prime ideal of A . With an aim to construct a ring with involution of this kind, we study the behavior of generalized derivations satisfying the algebraic identities involving prime ideals. As a consequence, currently existing results in this field are enhanced.

A Pair of Derivations on Prime Rings with Antiautomorphism

This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems.

Formula omitted]-graded identities of the Lie algebras U1

Let K be an infinite field of characteristic different from two and let U1 be the Lie algebra of the derivations of the algebra of Laurent polynomials K[t,t−1]. The algebra U1 admits a natural Z-grading. We provide a basis for the graded identities of U1 and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension ≤1, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra W1 of the derivations of the polynomial ring K[t]. The Z-graded identities for W1, in characteristic 0, were described in [8]. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in [8], and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra slq(K) with the Pauli gradings where q is a prime number.

Extensions of Some Known Algebraic and Combinatorial Identities

Extensions of Some Known Algebraic and Combinatorial Identities

On an identity of Sylvester

We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.

Elliptic modular graphs, eigenvalue equations and algebraic identities

We obtain eigenvalue equations satisfied by various elliptic modular graphs with five links where two of the vertices are unintegrated. Solving them leads to several nontrivial algebraic identities between these graphs.

On isomorphisms between ideals of Fourier algebras of finite abelian groups

On isomorphisms between ideals of Fourier algebras of finite abelian groups