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We show that in a prime nonassociative Novikov algebra every nonzero ideal is non-associative. We prove that Baer (and Andrunakievich) radical and the largest left quasiregular ideal 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. [ABSTRACT FROM AUTHOR]

Let A be a unital prime *-algebra with characteristic not 2 and with a projection e ≠ 0 , 1 . Suppose that d : A → Q s (A) is a map satisfying d (x y * z + z y * x 2) = d (x) y * z + x d (y) * z + x y * d (z) + d (z) y * x + z d (y) * x + z y * d (x) 2 for all x , y , z ∈ A , where Q s (A) denotes the symmetric Martindale algebra of quotients of A. It is shown that d is an additive triple derivation. Moreover, there exist a ∈ Q s (A) with a * = − a and an additive *-derivation δ : A → Q s (A) such that d (x) = a x + δ (x) for all x ∈ A . The analogous results for prime locally matrix *-algebras, prime *-algebras with nonzero socle, factor von Neumann algebras and standard operator *-algebras on Hilbert spaces are also described. [ABSTRACT FROM AUTHOR]

We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative semiprime algebra is semiprime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. [ABSTRACT FROM AUTHOR]

We deal with adjoint commutator and Jordan algebras of isotopes of prime strictly ( −1 , 1)-algebras. It is proved that a system of identities of the form [ x1, x2, x2, x3,..., x n] for n = 2 ,.... , 5 is discernible on isotopes of prime (−1, 1)-algebras. Also it is shown that adjoint Jordan algebras for suitable isotopes of prime (−1, 1)-algebras may possess distinct sets of identities. In particular, isotopes of a prime Jordan monster have different sets of identities in general. [ABSTRACT FROM AUTHOR]

We prove that every exceptional prime alternative algebra satisfies the Engel identity [[ x, y], y] = 0 of index 2. [ABSTRACT FROM AUTHOR]

We prove nilpotency of the alternator ideal of a finitely generated binary $(-1,1)$-algebra. An algebra is a binary $(-1,1)$-algebra if its every 2-generated subalgebra is an algebra of type $(-1,1)$. While proving the main theorem we obtain various consequences: a prime finitely generated binary $(-1,1)$-algebra is alternative; the Mikheev radical of an arbitrary binary $(-1,1)$-algebra coincides with the locally nilpotent radical; a simple binary $(-1,1)$-algebra is alternative; the radical of a free finitely generated binary $(-1,1)$-algebra is solvable. Moreover, from the main result we derive nilpotency of the radical of a finitely generated binary $(-1,1)$-algebra with an essential identity. [ABSTRACT FROM AUTHOR]

It is proved that a finitely generated prime monomial algebra either is primitive or it satisfies a polynomial identity or it is not semiprimitive. This confirms a conjecture of Bell and Smoktunowicz. [ABSTRACT FROM AUTHOR]