On the Lattice of Strong Radicals

AbstractIt is shown that the class of all strong radicals containing the prime radical is not a sublattice of the lattice of all radicals. This gives a negative answer to some questions of Sands and Puczylowski

On Herstein's Lie Map Conjectures, II

AbstractThe theory of functional identities is used to study derivations of Lie algebras arising from associative algebras. Definitive results are obtained modulo algebras of “low dimension.” In particular, Lie derivations of [K,K]/([K,K]∩Z), where K is the Lie algebra of skew elements of a prime algebra with involution and Z is its center, are described. This solves the last remaining open problem of Herstein on Lie derivations. For a simple algebra with involution the Lie algebra of all derivations of [K,K]/([K,K]∩Z) is thoroughly analyzed. Maps that act as derivations on arbitrary fixed polynomials are also discussed, and in particular a solution is given for Herstein's question concerning maps of K which act like a derivation on xm, m being a fixed odd integer

On graded polynomial identities with an antiautomorphism

AbstractLet G be a commutative monoid with cancellation and let R be a strongly G-graded associative algebra with finite G-grading and with antiautomorphism. Suppose that R satisfies a graded polynomial identity with antiautomorphism. We show that R is a PI algebra

Lie Isomorphisms in Prime Rings with Involution

AbstractLet R and R′ be prime rings with involutions of the first kind and with respective Lie subrings of skew elements K and K′. Furthermore assume (RC : C) ≠ 1, 4, 9, 16, 25, 64, where C is the extended centroid of R. It is shown that any Lie isomorphism of K onto K′ can be extended uniquely to an associative isomorphism of 〈K〉 onto 〈K′〉, where 〈K〉 and 〈K′〉 are respectively the associative subrings generated by K and K′

On Lie-Admissible Algebras Whose Commutator Lie Algebras Are Lie Subalgebras of Prime Associative Algebras

AbstractWe describe third power associative multiplications ∗ on noncentral Lie ideals of prime algebras and skew elements of prime algebras with involution provided that x∗y−y∗x=[x,y] for all x,y and the prime algebras in question do not satisfy polynomial identities of low degree. We also obtain necessary and sufficient conditions for these multiplications to be fourth power-associative or flexible

On Frobenius algebras and the quantum Yang-Baxter equation

It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya

Nonsingular CS-rings coincide with tight PP rings

AbstractIt is shown that if M is a self-generator right R-module, then M is non-M-singular and CS iff M is M-tight and End(MR) is a right PP ring. In particular, right nonsingular right CS-rings R are precisely right PP and right R-tight. As applications we show, among others, that for any domain R, RR2 is right CS if and only if R is two-sided Ore domain and two-sided 2-hereditary, giving answer to an open question known previously in special cases. As another application, we show that for a von Neumann regular ring R, the matrix ring Mn(R), n>1, is right weakly selfinjective if and only if R is right selfinjective