# Sur les dimensions des algèbres absolument valuées

- Authors
- Journal
- Journal of Algebra 0021-8693
- Publisher
- Elsevier
- Publication Date
- Volume
- 68
- Issue
- 2
- Identifiers
- DOI: 10.1016/0021-8693(81)90262-3
- Disciplines

## Abstract

Abstract Let A denote an arbitrary absolute valued real algebra. In [ Proc. Amer. Math. Soc. 11 (1960), 861–866.], Urbanik and Wright showed that if A is commutative, then A has dimension 1 or 2 and is isomorphic to R or C . In this paper we show that if in the previous result we replace the commutativity by the weaker assumption that it is flexible, then A is still finite dimensional, but in this case A has dimension 1, 2, 4 or 8 and is isotopic to R, C, H or C. Moreover, we consider various conditions on A which imply that it is finite dimensional (necessarily of dimension 1, 2, 4 or 8). In fact, we prove that it is the case when one of the following holds: (i) ( x, x, x) = 0 for all x ϵ A and there exists in A, a non-zero element, which commutes with all elements of A; (ii) there exists a non-zero element a in A such that a and a 2 commute with all elements of A and ( x, a, x) = 0 for all x ϵ A.

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