A Topological-Rough Algebraic Framework: Extending Approximations to Ring and Fields
Department of Mathematical, Open Educational College. Wasit, Iraq.
Research Article
Open Access Research Journal of Science and Technology, 2025, 15(02), 119-128.
Article DOI: 10.53022/oarjst.2025.15.2.0141
Publication history:
Received on 25 October 2025; revised on 04 December 2025; accepted on 07 December 2025
Abstract:
The important effort of Pawlak established Rough Set Theory to robust paradigm for handling uncertainty by defining approximation of concepts within an equivalence-based approximation space
. Subsequent research has explored the connections between rough sets, algebraic structures, and topology. Though, a unified framework that seamlessly integrates these three domains for complex algebraic structures like rings and fields remains largely undeveloped.
. Subsequent research has explored the connections between rough sets, algebraic structures, and topology. Though, a unified framework that seamlessly integrates these three domains for complex algebraic structures like rings and fields remains largely undeveloped.This paper introduce a novel hybrid structure, the Topological-Rough Algebraic Framework, which generalizes the concept of a topological approximation space to the realm of rings and fields. We construct an approximation space
) where the equivalence relation
is induce for Ideal
of a Ring
, partitioning to Ring into cossets. This partition naturally serves as a aim for a Topol
gy
to
, leading to the integrated triple
),
) where the equivalence relation
is induce for Ideal
of a Ring
, partitioning to Ring into cossets. This partition naturally serves as a aim for a Topol
gy
to
, leading to the integrated triple
),Within the proposed framework we investigate lower and upper approximation of subsets, characterizing the conditions under which a set is exact or rough. We further analyze key topological properties of the space
) establishing their relationships with the underlying algebraic properties of the ring and its ideals. The framework not only provides a theoretical generalization of existing models but also demonstrates practical applicability in coding theory, cryptographic analysis, and data mining. This work bridges a significant gap between algebraic structures, rough set theory, and general topology, offering a powerful new lens for mathematical analysis.
) establishing their relationships with the underlying algebraic properties of the ring and its ideals. The framework not only provides a theoretical generalization of existing models but also demonstrates practical applicability in coding theory, cryptographic analysis, and data mining. This work bridges a significant gap between algebraic structures, rough set theory, and general topology, offering a powerful new lens for mathematical analysis.
Keywords:
Rough Sets; Topological Approximation Space; Rings; Ideals; Coset topology; Algebraic-Structures; Data Analysis
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