Geodesic
Simple right-symmetric $(1,1)$-superalgebras
Algebra i logika, Tome 60 (2021) no. 2, pp. 166-175.

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It is proved that $2$-torsion-free prime right-symmetric superrings having a nontrivial idempotent and satisfying a superidentity $(x,y,z)+(-1)^{z(x+y)}\cdot (z,x,y)+(-1)^{x(y+z)}(y,z,x)=0$ are associative. As a consequence, every simple finite-dimensional $(1,1)$-superalgebra with semisimple even part over an algebraically closed field of characteristic $0$ is associative.
Keywords: right-symmetric ring, left-symmetric algebra, pre-Lie algebra, prime ring, Peirce decomposition, $(1,1)$-superalgebra. 
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A. P. Pozhidaev; I. P. Shestakov. Simple right-symmetric $(1,1)$-superalgebras. Algebra i logika, Tome 60 (2021) no. 2, pp. 166-175. http://geodesic.mathdoc.fr/item/AL_2021_60_2_a3/

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