Homomorphisms on Bicomplex Modules and Their Algebraic Properties

Authors: Neetu Singh

DOI: https://doi.org/10.37082/IJIRMPS.v14.i4.233186

Short DOI: https://doi.org/

Country: India

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Abstract: Modules over a commutative ring with two orthogonal central idempotents and numerous zero divisors are known as bicomplex modules. Consequently, their homomorphism theory is more comprehensive in terms of terminology, but it is structurally reducible to paired complex-linear algebra. This paper provides a self-contained algebraic treatment of bicomplex-module homomorphisms. A unique pair of complex-linear mappings is demonstrated for each homomorphism φ:M→N, and each module M is decomposed as M=e₁M₁⊕e₂M₂. Kernels, images, submodules, quotients, direct sums, exact sequences, endomorphisms, and automorphisms are characterised componentwise. Proofs for the correspondence theorem and the first, second, and third isomorphism theorems are provided in detail. Despite the fact that a module does not need to be free, every brief exact sequence divides because the bicomplex ring is isomorphic to a product of two fields. The minimum number of generators is the larger component dimension, and a finite module is free of rank r precisely when both component dimensions equal r. This results in a rank vector that is more informative than a singular rank. The simplistic concept of torsion is rendered degenerate by zero divisors, as each pure idempotent vector is destroyed by a nonzero scalar. Consequently, a regular-scalar concept is distinguished. Applications to algebraic operator models, invariant submodules, and bicomplex matrices are examined. Open problems pertain to the classification of constrained homomorphisms, homological invariants under additional structure, tensor products, and topological modules.

Keywords: bicomplex module; module homomorphism; idempotent decomposition; isomorphism theorem; exact sequence; free module; rank vector; zero divisor

Paper Id: 233186

Published On: 2026-07-28

Published In: Volume 14, Issue 4, July-August 2026

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