Simple right-symmetric $(1,1)$-superalgebras

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.