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All Journal Contemporary Mathematics and Applications (ConMathA) Semeton Mathematics Journal
Ambar, Jinan
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Materials Science & Nanotechnology Mathematics

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The Szeged Index and Padmakar-Ivan Index on the Zero-Divisor Graph of a Commutative Ring Ambar, Jinan; I Gede Adhitya Wisnu Wardhana; Abdurahim
Contemporary Mathematics and Applications (ConMathA) Vol. 7 No. 1 (2025)
Publisher : Universitas Airlangga

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20473/conmatha.v7i1.63517

Abstract

The zero-divisor graph of a commutative ring is a graph where the vertices represent the zero-divisors of the ring, and two distinct vertices are connected if their product equals zero. This study focuses on determining general formulas for the Szeged index and the Padmakar-Ivan index of the zero-divisor graph for specific commutative rings. The results show that for the first case of ring, the Szeged index is exactly half of the Padmakar-Ivan index. For the second case, the Szeged index is consistently greater than the Padmakar-Ivan index. These findings enhance the understanding of how the algebraic structure of rings influences the topological properties of their associated graphs.
Studi Keprimaan dalam Modul: Submodul Prima, Prima Lemah, Hampir Prima, dan n- Hampir Prima Ambar, Jinan; Afdhaluzzikri, Muhammad
Semeton Mathematics Journal Vol 1 No 2 (2024): Oktober
Publisher : Program Studi Matematika

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29303/semeton.v1i2.216

Abstract

Module theory, as a branch of abstract algebra, is a field of study that extends the concept of vector spaces into a more general framework, finding broad applications across various mathematical fields. The concept of prime numbers, initially abstracted by Dedekind in the form of prime ideals, underwent further development by mathematicians. Anderson generalized it into weak prime ideals, while Sharma developed the concept of nearly prime ideals. After Noether introduced the concept of modules, Dauns brought this primality notion into module theory under the name of prime submodules. Subsequently, Khashan generalized it into weak and nearly prime submodules. The latest development came from Azizi, who introduced the concept of nearly prime submodules into n-nearly prime submodules. In this context, this article discusses several key characteristics of prime submodules and their generalizations, providing a deep understanding of their fundamental structures and properties. Through this understanding, we can explore various applications of module theory in various mathematical contexts and delve into the complexities inherent in the study of primality and submodule structures within modules. The purpose of this research is to explain some structures and properties of prime submodules. This research uses the literature review method, which is by collecting information from various reading sources related to prime submodules and their generality. The result of this research is an explanation of some structures and properties of prime submodules and their examples.
Karakteristik Beberapa Submodul dari Suatu Modul Afdhaluzzikri, Muhammad; Ambar, Jinan
Semeton Mathematics Journal Vol 1 No 2 (2024): Oktober
Publisher : Program Studi Matematika

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29303/semeton.v1i2.217

Abstract

In the context of group cohomology, free modules are used to construct projective resolutions, while torsion modules play a significant role. In the theory of invariants, free modules describe all polynomial invariants, whereas torsion modules are relevant in the study of modular invariants. In elliptic curve cryptography, torsion points ensure security, with free modules used for arithmetic operations. The article also discusses the structure and properties of submodule types such as torsion, free, indecomposable, cyclic, and pure submodules, highlighting the importance of understanding the various applications of free and torsion modules.
Co-Authors Abdurahim, Abdurahim Afdhaluzzikri, Muhammad I Gede Adhitya Wisnu Wardhana