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Dupliquée d’une algèbre et le théorème d’Etherington. (Duplicate of an algebra and Etherington’s theorem). (French) Zbl 0734.17003
The authors prove that the duplicate algebra D(A) of an algebra A over a commutative field K is baric, a T-algebra or a genetic algebra, as soon as \(A^ 2\) has the same property. Moreover, the duplicate of each Bernstein algebra is genetic (if characteristic \(K\neq 2)\). Furthermore, if K is a commutative ring with unity, if the K-module \(A^ 2\) is projective and \(A^ 2=A\), then the derivation algebras of A and D(A) are isomorphic. Finally the authors make clear that, although \(D(A\otimes_ KB)\simeq D(A)\otimes_ KD(B)\) if D indicates the noncommutative duplicate, this isomorphism is no longer valid for a commutative duplicate.

MSC:
17A60 Structure theory for nonassociative algebras
17D92 Genetic algebras
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