On radicals of Novikov algebras

TL;DR

The paper advances radical theory for Novikov algebras by linking primeness, semiprimeness, and radical structures. It proves that in prime nonassociative Novikov algebras every nonzero ideal is non-associative, and that in finite dimensions the Baer radical and Andrunakievich radical coincide, with the left quasiregular radical agreeing with Baer under char $0$ or algebraically closed odd char. It further shows that finite-dimensional Novikov algebras are left quasiregular exactly when they are solvable, establishing a clear solvability-radical correspondence under the stated field conditions, and it demonstrates the nonexistence of a right quasiregular radical in finite dimensions. Collectively, these results clarify the structure of radicals in the Novikov setting and inform the classification of algebras in this variety. The findings have potential implications for the broader understanding of ideal structure and radical hierarchies in nonassociative algebra theory.

Abstract

We show that in a prime nonassociative Novikov algebra every nonzero ideal is non-associative. We prove that Baer (and Andrunakievich) radical and left quasiregular radical coincide in finite dimensional Novikov algebras over a field of characteristic 0 or algebraically closed field of odd characteristic. We show non-existence of right quasiregular radical in finite dimensional Novikov algebras.