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Now showing 1 - 5 of 14

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Matrices of the Zero Divisor Graphs of Classes of 3-Radical Zero Completely Primary Finite Rings
(SCIENCE MUNDI, 2024) Frank Omondi Ndago; Maurice Owino Oduor; Michael Onyango Ojiema
The study of finite completely primary rings through the zero divisor graphs, the unit groups and their associated matrices, and the automorphism groups have attracted much attention in the recent past. For the Galois ring R ′ and the 2-radical zero finite rings, the mentioned algebraic structures are well understood. Studies on the 3-radical zero finite rings have also been done for the unit groups and the zero divisor graphs Γ(R). However, the characterization of the matrices associated with these graphs has not been exhausted. It is well known that proper understanding of the classification of zero divisor graphs with diameter 2 and girth 3 can provide insights into the structure of commutative rings and their zero divisors. In this study, we consider a class of 3-radical zero completely primary finite rings whose diameter and girth are 2 and 3 respectively. We enhance the understanding of the structure of such rings by investigating their Adjacency, Laplacian and Distance matrices.
On the Structure of a Class of Galois Ring Module Idealization
(2024) Owino Maurice Oduor
Let Ro be a Galois ring and U is a finitely generated Ro− module. Consider an idealization of U expressed as R = Ro ⊕U endowed with a suitable multiplication. We explore the structure of R through its group of units R × and the graph of its zero divisors Γ(R). The study involves an investigation on the overarching interplay between the ring theoretical properties of R, the group theoretic properties of R × and the graph theoretic properties of Γ(R). Since R is a finite ring with identity, the convention that each element of R is either a unit or a zero divisor has been extensively used to drive the concept of classification of the elements of R. The units of R have been classified, the automorphisms of R have been determined and the zero divisors of R have been characterized.
The Wiener Index and Related Indices of the Zero Divisor Graphs of Certain Classes of Completely Primary Finite Rings
(Asian J. Math. Appl. (2025) 2025:11, 2025) Frank Omondi Ndago; Maurice Oduor Owino
Studies on zero divisor graphs of completely primary nite rings R have been extensively done from Galois rings to other rings whose sets of zero divisors Z(R) coincides with the Jacobson radical J (R). The studies have focused on graph geometric properties such as the girth, clique number, chromatic number and diameter among others. Some ndings are also evident on matrices of zero divisor graphs on certain classes of rings. The classes of completely primary nite rings considered in the various studies are square radical zero, cube radical zero and power four radical zero. In this paper we have advanced the study on zero divisor graph Γ(R) of completely primary nite rings by investigating the Wiener Index and its invariants such as the Average disorder number and Distance index. Further, we analyse the binding number and some bounds on the Zagreb indices of the rings satisfying the conditions (Z(R))3 = (0) and (Z(R))2 ≠ (0), (Z(R))4 = (0) and (Z(R))3 ≠ (0).
Zero divisor graphs of classes of five radical zero commutative completely primary finite rings
(Int. J. Nonlinear Anal. Appl. In Press, 1–10, 2024-05) Hezron Saka Were; Maurice Owino Oduor
This paper provides a characterization for zero divisor graphs of a completely primary finite ring R satisfying the conditions (Z (R))5 = (0) ; (Z (R))4 ̸= (0) where Z(R) is its subset of all zero divisors (including zero). This has been achieved through Anderson and Livingston’s zero divisor graphs by precisely determining the graph invariants, including diameter, girth and the binding number, and graph characteristics including completeness, connectedness and partiteness.
Numerical Investigation of Turbulent Fluid Flow Over a Porous Aerofoil Wing Design Within a Magnetic Field
(International Advanced Research Journal in Science, Engineering and Technology, 2024) Kirui G. K; Mukuna W. O; Oduor M. O
A mathematical model of turbulent fluid flow over a porous aerofoil wing design within a magnetic field is considered. The fluid flow was modelled using Navier stokes equations of conservation of momentum, energy and mass in cylindrical coordinates. The governing equations were then non-dimentionalized and gave rise to the non-dimensional parameters. Computational fluid dynamics (CFD) techniques was used to simulate the flow of air over a porous wing within a range of magnetic field strengths. Examinations of the effects of the magnetic field on key performance metrics such as lift, drag, and efficiency, as well as the overall flow structure of the wing was performed and found valuable insights into the use of porous aerofoil wings in the design of aircraft operating in high-magnetic field environments, such as those found in space or near the Earth's poles. Additionally, the outcomes of the research had wider implications for other domains investigating the impact of magnetic fields on fluid motion, such as in the design of magnetic resonance imaging systems or in the study of planetary motions. Aerofoil wings are an essential component of aircraft design, as they provide lift and enable flight. However, the flow of air over the wing is often turbulent, which can lead to decreased efficiency and performance. Porous aerofoil wings was proposed as a means of reducing turbulence, and the effects of such wings on fluid flow within a magnetic field have been thoroughly investigated. In this research, numerical investigation of the effects of a magnetic field on turbulent fluid flow over a porous aerofoil wing design was done. It is evident from the results that the primary velocities increase when the magnetic parameter was reduced. It was also found that the lift force increases when the Grashof number and Prandtl number decreases.