On involutions over amalgamated algebras along an ideal

Abstract

Let $A$ and $B$ be two associative rings, $I$ be a two-sided ideal of $B$, and $f \in Hom(A,B)$. In this paper, we study the involutions on amalgamated algebras. Further, we construct a specific type of involutions on $A \bowtie^fI$ named amalgamated involutions. The paper investigates the Hermitian and skew-Hermitian elements of $A \bowtie^f I$ and determines the sets $H(A \bowtie^f I)$ and $S(A \bowtie^f I)$ for amalgamated involutions. Moreover, the paper derives several identities that establish the commutativity of $A \bowtie^f I$ when $A$ is prime. This allows to construct non-prime rings in which these identities imply their commutativity.

Keywords

• Involution
• amalgamated algebra
• prime ring

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