On the extension of automorphisms of polynomial rings

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 4, pp. 911–916, July–August, 1995.     

Schreier varieties of algebras

Some necessary and sufficient conditions for a variety of algebras to possess the Schreier property are given. It is proved that the variety of algebras defined by the identity xx ² = 0is a Schreier variety. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 317–340, May–June, 1994.     

Centralizers in free Poisson algebras

We prove an analog of the Bergman Centralizer Theorem for free Poisson algebras over an arbitrary field of characteristic 0. Some open problems are formulated.

     

Generic, almost primitive and test elements of free Lie algebras

We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.

     

Free Akivis Algebras, Primitive Elements, and Hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved.The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed.Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.