Abstract:
The connections between semiprime
associative $Z_{2}$-graded algebras and Jordan superalgebras are
studied. It is proved that if an adjoint Jordan superalgebra
$B^{(+)_{s}}$ to an associative noncommutative $Z_{2}$-graded
semiprime superalgebra $B$ contains an ideal, consisted of odd
elements, then the center of algebra $B$ contains a nonzero ideal.
Besides, this ideal annihilates every commutator of the algebra
$B$. As a corollary we have that if a $Z_{2}$-graded algebra $B$
is just infinite then a Jordan superalgebra $B^{(+)_{s}}$ is just
infinite.
Keywords:associative algebras, Jordan superalgebras, just infinite algebras, semiprime algebras.