Maximal Ideals in a Bicomplex Algebra and Bicomplex Gelfand-Mazur Theorem

TL;DR

This paper investigates the structure of maximal ideals in bicomplex algebras, introduces bicomplex division algebras, and extends the Gelfand-Mazur theorem to this setting, revealing new algebraic properties.

Contribution

It characterizes maximal ideals in bicomplex algebras and generalizes the Gelfand-Mazur theorem for bicomplex division Banach algebras, a novel extension in algebra theory.

Findings

Kernel of a nonzero multiplicative BC-linear functional need not be maximal

Maximal ideals in bicomplex algebras are described explicitly

Gelfand-Mazur theorem is extended to bicomplex division Banach algebras

Abstract

In this paper we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand-Mazur theorem for the bicomplex division Banach algebra.