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SHAKIR ALI
Departments of Biotechnology, Faculty of Science, Jamia Hamdard (Hamdard University) Hamdard Nagar, New Delhi-110062, India.
Biochemistry, Genetics & Molecular Biology Chemical Engineering, Chemistry & Bioengineering Environmental Science Materials Science & Nanotechnology Mathematics Physics

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Reverse Homoderivations on (Semi)-prime Rings Ali, Shakir; Rafiquee, Naira Noor; Varshney, Vaishali; Dy, Outdom
Journal of the Indonesian Mathematical Society Vol. 31 No. 2 (2025): JUNE
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.v31i2.1867

Abstract

In this paper, we explore and examine a new class of maps known as reverse homoderivations. A reverse homoderivation refers to an additive map g defined on a ring T that satisfies the condition, g(ϑℓ)=g(ℓ)g(ϑ)+g(ℓ)ϑ+ℓg(ϑ), for all ϑ,ℓ∈T. We present various results that enhance our understanding of reverse homoderivations, including their existence in (semi)-prime rings and the behavior of rings when they satisfy certain functional identities. Some examples are provided to demonstrate the necessity of the constraints, while additional examples are given to clarify the concept of reverse homoderivations.
Commuting and Centralizing Maps on Modules Fitriani, Fitriani; Wijayanti, Indah Emilia; Faisol, Ahmad; Ali, Shakir
Science and Technology Indonesia Vol. 10 No. 3 (2025): July
Publisher : Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26554/sti.2025.10.3.690-697

Abstract

A ring is a mathematical structure composed of a set with two binary operations that follow certain axioms. One important function within a ring is the centralizing and commuting mapping, which has been extensively studied in recent decades. Commuting mappings are a special case of centralizing mappings. A module is a generalization of a ring. In this paper, we extend the concept of commuting mappings from ring to module structures. However, defining commuting mappings in modules presents a challenge, as multiplication is required for their definition, yet modules do not have this operation. Additionally, constructing nonzero centralizing and commuting mappings on modules is a nontrivial task. To address these challenges, we employ the concept of idealization as a framework for defining commuting mappings in modules. We also propose a method for constructing nonzero commuting mappings on modules by leveraging existing commuting mappings in rings. Specifically, if α is a commuting mapping on a ring T, then a corresponding commuting mapping α’ can be defined on the module by utilizing α. Moreover, we establish that the finite sum of commuting mappings is also a commuting mapping and that a linear combination of  commuting mappings is also a commuting mapping under certain conditions.
ON QUANTUM CODES CONSTRUCTION FROM CONSTACYCLIC CODES OVER THE RING I_q[u,v] / Ali, Shakir; Sharma, Pushpendra
Journal of the Indonesian Mathematical Society Vol. 30 No. 2 (2024): JULY
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.30.2.1587.139-159

Abstract

This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring $\mathfrak{S}$, which is defined as $\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangle$, where $\mathfrak{I}_q$ is a finite field of $q$ elements, $\alpha$ be the nonzero elements of the field $\mathfrak{I}_q$ and $q$ is a power of an odd prime $p$ such that $q=p^m, ~\textup{for}~ m \ge 1$. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring $\mathfrak{S}$ into a direct sum of constacyclic codes over $\mathfrak{I}_q$. We construct new and better quantum error-correcting codes over the ring $\mathfrak{S}$ (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).
CERTAIN TYPES OF DERIVATIONS IN RINGS: A SURVEY Ali, Shakir; Rafiquee, Naira Noor; Varshney, Vaishali
Journal of the Indonesian Mathematical Society Vol. 30 No. 2 (2024): JULY
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.30.2.1623.256-306

Abstract

In this overview article, we provide a historical account on derivations, Jordan derivations, (α, β)-derivations, left derivations, pre-derivations, homoderivations, nilpotent derivations, and other variants, drawing from the contributions of multiple researchers. Additionally, we delve into recent findings and suggest potential avenues for future investigation in this area. Furthermore, we offer pertinent examples to illustrate that the assumptions underlying various results are indeed necessary and not redundant.
Co-Authors Ahmad Faisol Dy, Outdom Fitriani Indah Emilia Wijayanti Rafiquee, Naira Noor Sharma, Pushpendra Varshney, Vaishali