Let A be an absolute valued algebra. In El-Mallah (El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 5 1: 39-49) we proved that, if A is algebraic with. an involution, then A is finite dimensional. This result had been generalized in El-Amin et al. (El-Amin, K., Ramirez, M. I., Rodriguez, A. (1997). Absolute. valued algebraic algebras are finite dimensional. J Algebra 195:295-307), by showing that the condition "algebraic" is sufficient for A to be finite dimensional. In the present paper we give a generalization of the concept "algebraic", which will be called "semi-algebraic", and prove that if A is semi-algebraic with an involution then A is finite dimensional. We give an example of an absolute valued algebra which is semi-algebraic and infinite dimensional. This example shows that the assumption "with an involution" cannot be removed in our result. [References: 5]
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