Abstract :

Idempotent Elements Those Satisfying The Condition 2 E E = Play A Critical Role In Understanding The Internal Structure Of Rings. While Extensively Studied In Commutative Ring Theory, Their Behavior In Non-commutative Rings Remains Comparatively Underexplored. This Paper Aims To Offer New Theoretical Perspectives On The Characterization And Functional Role Of Idempotent Elements In Non-commutative Algebraic Systems. We Investigate Conditions For The Existence And Uniqueness Of Idempotent In Various Classes Of Non-commutative Rings, Including Matrix Rings And Group Rings, And Examine Their Interaction With Ideals, Modules, And Ring Decompositions. Employing Algebraic And Ring-theoretic Methodologies, We Derive Several Novel Results On The Classification And Structure Of Idempotent, Supported By Illustrative Examples And Generalizations. A Key Contribution Is The Identification Of Specific Non Commutative Configurations Where Idempotent Behavior Diverges From Classical Commutative Patterns. Beyond Theoretical Interest, The Findings Have Practical Significance In Coding Theory And Cryptography. Idempotent Elements Are Instrumental In Constructing Orthogonal Codes, Generating Cyclic Sub Modules, And Designing Secure Algebraic Key Structures. The Results Enhance The Algebraic Toolkit Available For Building Robust Systems In Error Detection And Information Security. Overall, This Study Advances The Theoretical Understanding Of Idempotent In Non-commutative Contexts And Bridges The Gap Between Pure Algebra And Real-world Computational Applications. Keywords: Idempotent Element, Non Commutative Rings, Ring Theory, Algebraic Structures, Ring Decomposition, Coding Theory, Error-Correcting Codes Algebraic Cryptography And Ideal Theory

Published:

25-9-2025

Issue:

Vol. 25 No. 9 (2025)

Page Nos:

650-657

Section:

Articles

License:

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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