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2025 | Book

Algebra and Its Applications

ICAA-2023, Fez, Morocco, July 12–15

Editors: Manoj Kumar Patel, Mohammad Ashraf, Najib Mahdou, Hwankoo Kim

Publisher: Springer Nature Singapore

Book Series : Springer Proceedings in Mathematics & Statistics

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About this book

This volume contains selected chapters on algebra and related topics presented at the International Conference on Algebra and its Applications, held at the Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco, from 12–15 July 2023, held in honour of Prof. Ayman Badawi and Prof. Abdelmoujib Benkirane. It contains a cross-section of topics in algebra and its applications which contribute to the development of pure and applied algebra. Chapters in the book focus on modern trends and techniques in various branches of pure and applied algebra and highlight their applications in several other branches of mathematics like coding theory, cryptography and graph theory. Covering a broad range of topics in pure and applied algebra, the book will be useful to a wide spectrum of researchers and graduate students in mathematics.

Table of Contents

Frontmatter
Strongly Hopfian Rings of Formal Power Series and Power Series Armendariz Rings in Infinitely Many Variables
Abstract
In this paper, we study the strongly Hopfian property in the ring of formal power series in infinitely many variables over chained rings, nonzero characteristic rings and SFT-rings. We investigate the notion of power series Armendariz rings in infinitely many variables and its application to the strongly Hopfian property.
Ali Benhissi
Some Results on S-VSFT Commutative Rings
Abstract
A commutative ring A with identity is called S-VSFT (S-very strong finite type), where \(S\subseteq A\) is a given multiplicative set, if for each ideal I of A, there exist \(s\in S\), \(k\ge 1\) and a finitely generated ideal \(F\subseteq I\) such that \(sI^k\subseteq F\) (in this case I is called an S-VSFT ideal). In this paper, we give several generalizations of some results on VSFT rings. For instance, the ring A is S-VSFT if and only if each prime ideal of A is S-VSFT. Besides, we get some new results concerning the VSFT rings. In fact, we show that the ring A is S-VSFT if and only if the ring \(A(+)M\) is \(S(+)M\)-VSFT, where M is an A-module and \(A(+)M\) is idealization of M in A. Which turns out that A is VSFT if and only if \(A(+)M\) is VSFT. Many other results are also established.
Abdelamir Dabbabi, Ali Benhissi
Zappa–Szép Product of Right Admissible Groups
Abstract
In this paper, we have proved that if two groups are right admissible, then their Zappa–Szép product is also right admissible.
Ratan Lal, Ramjash, Vipul Kakkar
Binary Quantum Codes from Cyclic Codes over
Abstract
Let \(\mathfrak {R}_{s_{1},s_{2}}=\mathbb Z_4+s_{1}\mathbb Z_4+s_{2}\mathbb Z_4+s_{1}s_{2}\mathbb {Z}_4\) be a ring where \(s_{1}^2=s_{1}\), \(s_{2}^2=s_{2}\) and \(s_{1}s_{2}=s_{2}s_{1}\). We obtain the generating set of cyclic codes over the ring \(\mathfrak {R}_{s_{1},s_{2}}\) and a new \(\mathbb Z_2\)-linear isometry \(\phi \) from \(\mathfrak {R}_{s_{1},s_{2}}^{n}\) to \(\mathbb Z^{8n}_2\) is constructed. Further, we obtain the structure of dual of cyclic codes over the ring \(\mathfrak {R}_{s_{1},s_{2}}.\) By utilizing the images of the Gray map \(\phi ,\) some binary quantum error-correcting codes are constructed.
Mohammad Ashraf, Washiqur Rehman, Naim Khan, Ghulam Mohammad, Mohd Asim
On the Genus and Crosscap of the Extended Sum Annihilating-Ideal Graph of Commutative Rings
Abstract
In the context of a commutative ring with unity, denoted as \(\mathcal {S}\), and its associated set of annihilating ideals \(A(\mathcal {S})\), there exists a graph known as the extended sum annihilating-ideal graph, denoted as \(AG_\varOmega (\mathcal {S})\). This graph has its vertex set derived from the set \(A(\mathcal {S})^*\), and it exhibits a specific pattern of connections between its vertices. More precisely, two distinct vertices, referred to as \(\Im _1\) and \(\Im _2\), are linked by an edge if and only if one of the following conditions holds: either \(\Im _1\Im _2 = 0\) or \(\Im _1 + \Im _2 \in A(\mathcal {S})\). In the following research paper, we delve into the classification of Artinian commutative rings, denoted as \(\mathcal {S}\), with a particular focus on those where the extended sum annihilating-ideal graph takes on one of three distinct forms: a double toroidal graph, a projective plane, or a Klein bottle.
Mohd Nazim, Nadeem ur Rehman, Cihat Abdioğlu, Shabir Ahmad Mir, Nazim
Investigation of Ideal and Spectrum of Different Products on Matrices
Abstract
In this article, we discuss different types of products on matrices, like the Hadamard/Schur product and the Jordan product; and study their algebraic structure. Furthermore, our investigation extends to the examination of ideal and spectrum of these distinct matrix products.
Pragneshkumar R. Makwana
On Spectrum of Neighbourhood Corona Product of Signed Graphs
Abstract
Given two signed graphs \(\varGamma _1\) with node set \(\{u_1,u_2,\ldots ,u_n\}\) and \(\varGamma _2\), the neighbourhood corona, \(\varGamma _1*\varGamma _2\) is the signed graph obtained by taking one copy of \(\varGamma _1\) and \(n_1\) copies of \(\varGamma _2\), and joining every neighbour of the ith node with each node of the ith copy of \(\varGamma _2\) by a new signed edge. This paper will determine the condition for \(\varGamma _1*\varGamma _2\) to be balanced. We also determine the adjacency spectrum of \(\varGamma _1*\varGamma _2\) for arbitrary \(\varGamma _1\) and \(\varGamma _2\), and Laplacian and signless Laplacian spectrum of \(\varGamma _1*\varGamma _2\) for regular \(\varGamma _1\) and arbitrary \(\varGamma _2\), in terms of the corresponding spectrum of \(\varGamma _1\) and \(\varGamma _2\).
Bishal Sonar, Satyam Guragain, Ravi Srivastava
A Key Exchange Protocol Based on Quasidirect Product of Groups
Abstract
In this paper, we give the notion of key exchange protocol based on quasidirect product and transassociant and give some examples based on them.
Sushmita Kushwaha, Akhilesh Chandra Yadav, Varun Kumar
On k-Facile Perfect Numbers
Abstract
For a positive integer n, let \(\sigma (n)\) denote the sum of all positive divisors of n. Then n is said to be a k-facile perfect number if \(\sigma (n)=2n+d_1d_2\cdots d_k\), where \(1<d_1, d_2,\dots ,d_k<n\) are distinct divisors of n. This paper characterizes k-facile perfect numbers and establishes their relationships with other special numbers.
Flora Jeba S, Anirban Roy, Manjil P. Saikia
On the Role of the Fibonacci Matrix as Key in Modified ECC
Abstract
In this paper, we propose a modified cryptography based on the application of recursive matrices as key elements in ECC and ElGamal techniques. For encryption, we consider mapping analogous to affine Hill cipher, where a plaintext has been constructed by the points corresponding to letters on elliptic curves. Our key element for the proposed cryptography is based on the generalized Fibonacci matrices that make a large keyspace and its security strength is based on EC-DLP which is a hard problem in number theory.
Munesh Kumari, Jagmohan Tanti
On the Relation of Pure Projective Modules with Other Modules
Abstract
In this paper, our study aims to demonstrate certain relationships among finitely presented, projective, free, and pure projective modules. This article provides a comprehensive review of the literature, including illustrative examples, some theorems, and propositions.
Pooja Gupta, Ratnesh Kumar Mishra
On Lie Ideals with Generalized Homoderivations in Prime Rings
Abstract
Let \(\mathfrak {R}\) be a ring and \(\mathfrak {L}\) be a square closed Lie ideal of \(\mathfrak {R}\). A map \(H: \mathfrak {R}\rightarrow \mathfrak {R}\) is called a generalized homoderivation associated with homoderivation h if \(H(\varkappa \vartheta ) = H(\varkappa )h(\vartheta ) + H(\varkappa )\vartheta + \varkappa h(\vartheta )\) is fulfilled for all \(\varkappa ,\vartheta \in \mathfrak {R}\). The main objective of this paper is to examine the following identities: (1) \(\varkappa H(\vartheta ) \pm \varkappa \vartheta \in \mathfrak {Z}(\mathfrak {R})\), (2) \(\varkappa H(\vartheta ) \pm \vartheta \varkappa \in \mathfrak {Z}(\mathfrak {R})\), (3) \(\varkappa H(\vartheta ) \pm [\varkappa ,\vartheta ] \in \mathfrak {Z}(\mathfrak {R})\), (4) \(H(\vartheta )\varkappa \pm [\varkappa ,\vartheta ] \in \mathfrak {Z}(\mathfrak {R})\), (5) \([H(\varkappa ),\vartheta ] \pm \varkappa \vartheta \in \mathfrak {Z}(\mathfrak {R})\), (6) \([H(\varkappa ),\vartheta ] \pm \vartheta \varkappa \in \mathfrak {Z}(\mathfrak {R})\) for all \(\varkappa ,\vartheta \in \mathfrak {L}\) and prove that \(\mathfrak {L}\subseteq \mathfrak {Z}(\mathfrak {R})\) and \(\mathfrak {L}\) is commutative or \(h(\mathfrak {L}) = (0)\).
Wasim Ahmed, Muzibur Rahman Mozumder
Salted Cubic Unbalanced Oil and Vinegar Digital Signature Scheme
Abstract
In this paper, we propose a salted cubic unbalanced oil-vinegar (SCUOV) signature scheme. The SCUOV signature scheme relies on the difficulty of solving the multivariate cubic (MC) equation over the finite field \(\mathbb {F}_{p^n}\) where p-prime and \(n\in \mathbb {N}\). For the construction of SCUOV we utilize the step-wise iteration method which effectively helps in minimizing the public key size and signature size. We prove that our scheme exhibit short signature size compare to existing unbalanced oil-vinegar signature scheme with maintaining same security level. Additionally, we prove that SCUOV scheme shows resistant against Direct attack and Existential Unforgeability under Chosen Message Attack (EUF-CMA).
Ashok Ji Gupta, Satish Kumar, Swapna Kumar Biswal
A Note on Regular Qclean Rings
Abstract
Ashrafi and Nasibi introduced the notion of r-clean rings, where each element of ring is the sum of an idempotent and von Neumann regular element (an element \(x\in R\) is a von-neumann regular if there exists \(h\in R\) such that \(xhx=x\)). Motivated by this structure, we introduce here the notion of regular qclean ring by replacing idempotent element by q-potent element in r-clean ring. In this article, we focus to study the fundamental properties of regular qclean rings and also discussed some properties of matrix rings in context of regular qclean rings. Apart from this, we find some conditions on ring so that the ring of ideal extension is regular qclean.
R. G. Ghumde, M. K. Patel
One Type of Finitely Generated Minimal Linear Code Over the Ring
Abstract
In this paper we find one type of finitely generated minimal linear code over the ring \(\mathbb {Z}_{2n}\). We show that if the components of the vectors belongs to the ideal \(<n>\) then with a given conditions the module generated by these vectors must be minimal linear code.
Biplab Chatterjee, Ratnesh Kumar Mishra
Reductions and Sally Module
Abstract
In this paper, we study properties of multiplicity of the fiber cone of modules and the Sally module by using the theory of reductions of modules. We extend several results of the reductions of ideals to the reductions of modules and examples.
Priti Singh, Sharvan Kumar
The -labeling on Decomposition of Complete Bipartite Graphs
Abstract
In this paper, we study \(\varphi \)-labeling \(G_\varphi \) on decomposition of complete bipartite graph by special case of star graph. We prove that, if a complete bipartite graph G is isomorphic to its subgraph \(G_\varphi \), then for any \(K^j_i\)-copies of a complete bipartite graph of G has an \(\varphi \)-labeling. Finally we proved that, the decomposition of complete bipartite graph G into \(K^j_i\)-copies of the subgraph \(G_\varphi \) has a \(\varphi \)-labeling, where the total number of edges \(|E(G_\varphi )|\) of the \(1^{st}\)-\(K^j_i\)-copies is strictly less than the \(|E(G_\varphi )|\) of the \(n\textrm{th}\)-\(K^j_i\)-copies.
Aronthung S. Odyuo, M. K. Patel
A Finite Non-abelian SR-Group
Abstract
In this paper, we introduce a finite non-abelian group named the SR-group \(G_{2n} \) and prove the existence of such a group. We prove the converse of Lagrange’s theorem for SR-group \(G_{2n} \) and deduce the existence of some subgroups of symmetric group \(S_{2n} \). With the help of the SR-group, we prove that every SR-group is isomorphic to a proper subgroup of another SR-group. As a consequence of this we declare the existence of a permutation group of every even order which is isomorphic to a proper subgroup of some SR-group. Finally, we give an example to find a subgroup of symmetric group \(S_{24} \).
Rajesh Kumar, Subhash Chandra Singh
Sedenion Algebra of the k-Mersenne and k-Mersenne-Lucas Numbers
Abstract
In this paper, we study and investigate the k-Mersenne and k-Mersenne-Lucas sedenions. We give their algebraic properties in closed form and present some well-known identities such as Catalan’s identity, d’Ocagne’s identity, Vajda’s identity, etc. Moreover, for these sedenions we present ordinary and exponential generating functions and series sum formulas. In particular for \(k=1\), the results are presented for Mersenne and Mersenne-Lucas sedenions.
Kalika Prasad, Munesh Kumari
Graded Uniformly S-Noetherian Modules
Abstract
In this paper, we introduce the concept of G-graded uniformly S-Noetherian module as a generalization of uniformly S-Noetherian module and investigate some properties of this new class of modules. Our main aim is to characterize G-graded uniformly S-Noetherian modules in terms of uniformly S-Noetherian modules.
Ajim Uddin Ansari, Surya Prakash
A Note on Differential Identities in -Prime Ring with Generalized Derivations
Abstract
In this paper, we establish commutativity theorems for a \(\sigma \)-prime ring with an involution, where generalized derivations adhere to specific differential identities. We have also derived familiar results regarding the commutativity of prime rings. Finally, we offer an example to demonstrate that the condition presumed in our results is essential.
Md Arshad Madni, Muzibur Rahman Mozumder
Iso M-Artinian (Noetherian) Modules and Rings
Abstract
An Iso M-Artinian (Noetherian) modules are the generalization of an iso Artinian (Noetherian) modules, which is described as every descending (ascending) chain of M-cyclic submodules terminates with respect to isomorphism i.e., \(g_{n}(M)\cong g_{n+1}(M)\) for all \(n\ge k\) after a finite index \(k\in \mathbb {N}\), where \(g_{k}\in End (M)\). We proved that if M be an isosimple injective R-module and N be an iso M-Artinian (Noetherian), then \(N\oplus M\) is an iso \((N\oplus M\))-Artinian (Noetherian) R-module. For a ring R, we define an iso R-Artinian (Noetherian) ring, if it is an iso M-Artinian (Noetherian) module where M=\(_RR\). Also we proved that for a commutative iso R-Artinian ring R, if every non-zero ideal of R possesses an idempotent element, then R is Noetherian.
Himangshu Chakraborty, Manoj Kumar Patel
On Hamming Distance and RT Distance of a Class of Constacyclic Codes over 
Abstract
Assume that \(p^n \equiv 3\pmod 4\) and \(\alpha =\alpha _0+u\alpha _1+\cdots +u^{t-1}\alpha _{t-1}\) is Type 1 unit in \(\mathbb {F}_{p^n}+u\mathbb {F}_{p^n}+\cdots +u^{t-1}\mathbb {F}_{p^n} (u^t=0),\) where p is an odd prime, n is a positive integer and \(\alpha _0,\alpha _1,\ldots ,\alpha _{t-1} \in \mathbb {F}_{p^n}, \alpha _0\ne 0, \alpha _1\ne 0.\) In this paper, for \(p^n \equiv 3\pmod 4\), we determine Hamming distances and RT distances of all Type 1 \(\alpha \)-constacyclic codes of length \(4p^s\) over the chain ring \(\mathbb {F}_{p^n}+u\mathbb {F}_{p^n}+\cdots +u^{k-1}\mathbb {F}_{p^n}\) with the help of generator polynomials of these codes and their duals.
Saroj Rani
Uniform Coherence in Amalgamated Algebra Along an Ideal
Abstract
The concept of uniform coherence was first introduced by Soublin and has since been extensively explored by numerous researchers. A ring A is defined as uniformly coherent if there exists a function \(\phi : \mathbb {N} \rightarrow \mathbb {N}\), where \(\mathbb {N}\) represents the set of natural numbers, such that for every \(n \in \mathbb {N}\) and each nonzero homomorphism \(f : A^n \rightarrow A\), the kernel \(\ker (f)\) can be generated by \(\phi (n)\) elements. In this paper, we investigate how the property of uniform coherence is transferred to commutative ring extensions. We establish necessary and sufficient conditions for rings such as \(A\bowtie ^{f}J\), \(A\bowtie I\), and \(A\propto E\) to possess uniform coherence, covering various classes of ideals and A-modules. Our study contributes to the development of new classes of rings that meet this criterion, and we also introduce new families of rings that demonstrate the distinctiveness of the categories of Noetherian and uniformly coherent rings.
Karima Alaoui Ismaili, Hwankoo Kim, Najib Mahdou
Distance Laplacian Spectrum and Energy of Commuting Conjugacy Class Graphs
Abstract
Let G be a finite group and \(a^G\) be the conjugacy class of \(a\in G\). The commuting conjugacy class graph of G is a simple undirected graph, denoted by \(\Gamma _G\), whose vertex set is \(\{a^G : a \in G\}\) and two distinct vertices \(a^G\) and \(b^G\) are adjacent if there exist \(x\in a^G\) and \(y\in b^G\) such that x and y commute. In this paper, we compute distance Laplacian spectrum and energy of \(\Gamma _G\) if G is isomorphic to \(D_{2n}\) (dihedral group), \(T_{4n}\) (dicyclic group), \(U_{(n,m)}=\langle x,y:~x^{2n}=y^m=1,~x^{-1}yx=y^{-1}\rangle \), \(V_{8n}=\langle x,y:~x^{2n}=y^4=1,~yx=x^{-1}y^{-1},~y^{-1}x=x^{-1}y \rangle \) and \(SD_{8n}=\langle x,y:~x^{4n}=y^2=1,yxy=x^{2n-1}\rangle \). We also consider finite groups whose central quotient is isomorphic to \(\mathbb {Z}_p \times \mathbb {Z}_p\) (for any prime p) or \(D_{2n}\). Our computations reveal that the commuting conjugacy class graphs of the above mentioned groups are distance Laplacian integral.
Firdous Ee Jannat, Rajat Kanti Nath, Shariefuddin Pirzada
Nonlinear -Lie Higher Derivations of Unital -Algebras
Abstract
Let \(\mathcal {A}\) be a unital \(*\)-algebra and \(\mathcal {L}=\{L_n\}_{n\in \textrm{N}}\) be a nonlinear \(*\)-Lie higher derivation on \(\mathcal {A}.\) In the present paper, it is shown that under some appropriate assumptions \(\mathcal {L}\) is proper, that is, for each \(n\in \textrm{N},\) \(L_n:\mathcal {A}\rightarrow \mathcal {A}\) has the form \(L_n=d_n+\tau _n,\) where \(\{d_n\}_{n\in \textrm{N} }\) is an additive \(*\)-higher derivation on \(\mathcal {A}\) and \(\{\tau _n\}_{n\in \textrm{N} }\) is a family of mappings \(\tau _n:\mathcal {A}\rightarrow \mathcal {Z(A)}\) such that \(\tau _n([x,y])=0\) for all \(x,y\in \mathcal {A},\) \(n\in \textrm{N}\).
Mohammad Ashraf, Jehad Jumah Al Jaraden, Mohammad Afajal Ansari, Md Shamim Akhter
Rings in Which Every Ideal is S-Weakly Prime Ideal
Abstract
In this paper, we introduce and study the notion of rings in which every ideal is \(S-\) weakly prime. We next study the possible transfer of the above property in the direct product, homomorphic image, localization, trivial ring extensions, and amalgamated algebra along an ideal.
Chahrazade Bakkari, Rachid Hachache
On -V-Modules
Abstract
A ring R is called a (left) \(\pi \)-V-ring if, for every simple (left) R-module M, the injective hull E(M) of M is of finite length. The main goal of this paper is to introduce and study the notion of a (left) \(\pi \)-V-module. First, we define a radical \(G^R\) on the category \({{\,\textrm{Mod}\,}}(R)\) of R-modules. An R-module is called a \(\pi \)-V-module if \(G^R(N)=0\) for any module \(N\in \sigma [M]\). We prove that a (left) module M over a ring R is a \(\pi \)-V-module if and only if M is co-noetherian and co-artinian. As an application, we deduce that a ring R is a (left) \(\pi \)-V-ring if and only if R is a (left) co-noetherian and co-artinian ring.
A. Ait Ouahi, S. Bouchiba, Y. Najem
On a Generalization of -Modules
Abstract
A module M is called a \(C_{41}\)-module if whenever A is a nonsingular submodule of M and B is a direct summand of M with A isomorphic to B and \(A\cap B = 0\), then A is a direct summand of M. This concept is a generalization of the notion of \(C_4\)-modules. We show that direct sums of two \(C_{41}\)-modules do not inherit the property, in general. The class of rings R for which any arbitrary direct sum of \(C_{41}\)-modules is also a \(C_{41}\)-module is shown to be exactly that of right t-semisimple rings. We proved that the class of rings R for which every \(C_{41}\)-module satisfies the \(C_4\)-condition is precisely that of right SI-rings. We also investigate the rings R, whose all finitely generated free R-modules satisfy the \(C_{41}\)-condition.
Abdoul Djibril Diallo, Papa Cheikhou Diop, Farid Kourki, Rachid Tribak
Metadata
Title
Algebra and Its Applications
Editors
Manoj Kumar Patel
Mohammad Ashraf
Najib Mahdou
Hwankoo Kim
Copyright Year
2025
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-9767-98-4
Print ISBN
978-981-9767-97-7
DOI
https://doi.org/10.1007/978-981-97-6798-4

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