# zbMATH — the first resource for mathematics

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although $$D(A\otimes_ KB)\simeq D(A)\otimes_ KD(B)$$ if D indicates the noncommutative duplicate, this isomorphism is no longer valid for a commutative duplicate.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
Full Text:
##### References:
 [1] Abraham, V.M., A note on train algebras, Proc. Edinburgh math. soc., 20, 2, 53-58, (1976) · Zbl 0361.17007 [2] Boers, A.H., Duplication of algebras, Indag. math., 44, 121-125, (1982) · Zbl 0488.17002 [3] Boers, A.H., Duplication of algebras II, Indag. math., 50, 235-244, (1988) · Zbl 0661.17004 [4] Bourbaki, N., Algèbre I, (1970), Hermann Paris, chapitres 1-3 [5] Costa, R.C.F., On the derivation algebra of zygotic algebras for polyploidy with multiple alleles, Bol. soc. brasil. mat., 14, 1, 63-80, (1983) · Zbl 0575.17014 [6] Eilenberg, S., Extensions of general algebras, Ann. soc. polon. math., 21, 125-134, (1948) · Zbl 0031.34303 [7] Etherington, I.M.H., Duplication of linear algebras, Proc. Edinburgh math. soc., 6, 2, 222-230, (1941) · Zbl 0061.05302 [8] Holgate, P., Genetic algebras satisfying Bernstein’s stationarity principle, J. London math. soc., 9, 2, 613-623, (1975) · Zbl 0365.92025 [9] Micali, A., Dérivations dans LES algèbres gamétiques III, Linear algebra appl., 113, 79-99, (1989) · Zbl 0662.17018 [10] Micali, A.; Ouattara, M., Algèbres de Jordan génétiques, colloque de Zaragoza, Algebras, groups and geometries, (13-14 Avril 1989), à paraître [11] Odoni, R.W.K.; Stratton, A.E., Structure of Bernstein algebras, Algèbres Génétiques, 38, 117-125, (1989), Montpellier · Zbl 0754.17028 [12] Ouattara, M., Algèbres de Jordan et algèbres Génétiques, Cahiers math., 37, (1988), Montpellier [13] Peresi, L.A., A note on duplication of algebras, Linear algebra appl., 104, 65-69, (1988) · Zbl 0651.17013 [14] Schafer, R.D., Structure of genetic algebras, Amer. J. math., 71, 121-135, (1949) · Zbl 0034.02004 [15] Schafer, R.D., An introduction to nonassociative algebras, (1966), Academic New York · Zbl 0145.25601 [16] Walcher, S., Bernstein algebras which are Jordan algebras, Arch. math., 50, 218-222, (1988) · Zbl 0617.17013 [17] Wörz-Busekros, A., Bernstein algebras, Arch. math., 48, 338-398, (1987) · Zbl 0597.17014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.