Some homomorphisms on the ring of Banach topological algebras

Abstract

Ring homomorphisms are structure-preserving mappings between rings that are fundamental in abstract algebra. This paper explores ring homomorphisms and related concepts in ring theory. We introduce key definitions including ring homomorphisms, isomorphisms, and automorphisms. Properties of n-homomorphisms between complex algebras are presented, focusing on multiplicativity and stability. We then study homomorphisms on Fréchet algebras, deriving an inequality bounding the modulus of A-module homomorphisms where A is a unital Fréchet algebra. The continuity and boundedness of the modulus  are analyzed under various conditions. Further inequalities are established for the modulus of homomorphisms from a Fréchet algebra to a Banach algebra. The automatic continuity of homomorphisms from Fréchet algebras with bounded approximate identities into Banach algebras is demonstrated. The paper ends with summarizing the main results on continuity and boundedness of homomorphism moduli between algebraic structures in functional analysis. The theoretical development increases understanding of structure preservation for rings and algebras equipped with topological vector space structures..