Novikov $ ��_{2} $ -Graded Algebras with an Associative 0-Component

Abstract

In 1974 Kharchenko proved that if a \( 0 \)-component of an \( n \)-graded associative algebra is PI then this algebra is PI. In the Novikov algebras of characteristic 0 the existence of a polynomial identity is equivalent to the solvability of the commutator ideal. We study a \( 𝕑_{2} \)-graded Novikov algebra \( N=A+M \) and prove that if the characteristic of the basic field is not 2 or 3 and its 0-component \( A \) is associative or Lie-nilpotent of index 3 then the commutator ideal \( [N,N] \) is solvable.