Author: Chinibas Mahato
DOI Link: https://doi.org/10.70798/Bijmrd/03070025
Abstract: Boolean rings and matrix rings represent two distinct yet foundational structures in ring theory, each contributing uniquely to both theoretical and applied mathematics. Boolean rings, characterised by idempotent elements and characteristic two, offer a categorical equivalence to Boolean algebras and model logical operations algebraically. Their structure is inherently simple yet powerful, providing critical insights in digital logic design, error-correcting codes, and cryptographic systems. On the other hand, matrix rings generalise classical linear algebra by encapsulating higher-dimensional operations over arbitrary rings. They reveal rich non-commutative behaviour, ideal structures, and fundamental decomposition theorems such as Wedderburn’s. This paper explores original results about idempotent sums, zero-divisors, and quotient properties in Boolean rings, as well as structural theorems in matrix rings including trace-rank equivalence, invertibility, and ideal generation. Illustrative examples and diagrams support key identities, making abstract properties more tangible. The interplay between these rings demonstrates the versatility of ring theory in bridging abstract algebra, logic, and linear transformation frameworks. Through these investigations, this work contributes novel perspectives and self-contained theorems, advancing the understanding of these classical algebraic objects in a modern context.
Keywords: Boolean Ring, Matrix Ring, Idempotent Element, Non- Commutativity, Ring Theory, Zero-Divisor.
Page No: 221-238