In this study, we explore the properties of pre-Hilbert and absolute valued algebras, shedding light on their fundamental characteristics. A real algebra is considered a pre-Hilbert algebra when its norm is derived from an inner product. On the other hand, absolute valued algebras are those whose norms satisfy the equality condition ???????? = ???? ???? for all ????,???? ∈????. We investigate the relationships between these algebraic structures, and we extend Rodriguez's theorem to more general scenarios.
Specifically, we demonstrate that if a two-dimensional real algebra is considered, it can be isomorphic to a new class of two-dimensional pre-Hilbert algebras. Furthermore, when dealing with algebraic algebras containing nonzero central idempotents, we establish equivalences between the flexibility of the algebra, its degree, and its orthogonality properties within certain vector spaces.
This research contributes to a deeper understanding of the algebraic structures and their interplay, providing insights into the fundamental properties of pre-Hilbert and absolute valued algebras