On involutions over amalgamated algebras along an ideal

Year 2025, Volume: 38 Issue: 38, 197 - 211, 14.07.2025

Brahim Boudine , Mohammed Zerra

https://doi.org/10.24330/ieja.1611885

https://izlik.org/JA53MF57DT

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

References

• S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J., 23(1) (2016), 9-14.
• P. Charpin, S. Mesnager and S. Sarkar, Involutions over the Galois field ${\mathbb F} _ {2^{n}} $, IEEE Trans. Inform. Theory, 62(4) (2016), 2266-2276.
• M. D'Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl., 6(3) (2007), 443-459.
• M. D'Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat., 45(2) (2007), 241-252.
• A. Ebadian and A. Jabbari, $C^{\ast}$-algebras defined by amalgamated duplication of $C^{\ast}$-algebras, J. Algebra Appl., 20(2) (2021), 2150019 (15 pp).
• A. El Khalfi, H. Kim and N. Mahdou, Amalgamation extension in commutative ring theory: a survey, Moroc. J. Algebra Geom. Appl., 1(1) (2022), 139-182.
• M. A. Idrissi and L. Oukhtite, Derivations over amalgamated algebras along an ideal, Comm. Algebra, 48(3) (2020), 1224-1230.
• H. Javanshiri and M. Nemati, Amalgamated duplication of the Banach algebra $\mathfrak{A}$ along a $\mathfrak{A}$-bimodule $\mathcal{A}$, J. Algebra Appl., 17(9) (2018), 1850169 (21 pp).
• G. Luo, X. Cao and S. Mesnager, Several new classes of self-dual bent functions derived from involutions, Cryptogr. Commun., 11(6) (2019), 1261-1273.
• A. Mamouni, L. Oukhtite and M. Zerra, Certain algebraic identities on prime rings with involution, Comm. Algebra, 49(7) (2021), 2976-2986.
• S. Mesnager, M. Yuan and D. Zheng, More about the corpus of involutions from two-to-one mappings and related cryptographic S-boxes, IEEE Trans. Inform. Theory, 69(2) (2023), 1315-1327.
• B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Commutativity theorems in rings with involution, Comm. Algebra, 45(2) (2017), 698-708.