The Kothe Radical in Novikov Algebras

We study the Kothe radical (the upper nil radical) in the variety of Novikov algebras. We prove that the class of weakly right nilpotent Novikov algebras coincides with the class of r-nil algebras and forms a radical class. We establish that the Kothe radical is a hereditary radical in the variety of Novikov algebras. Furthermore, we obtain a structural description of the Kothe radical as the intersection of ideals whose corresponding quotient algebras are prime K-semisimple algebras. Finally, we construct an example of a prime r-nil Novikov algebra over a field of characteristic p>0, which implies that the Baer radical is strictly contained in the Kothe radical in this case.