Combinatorial extensions of radical, nil, and locally nilpotent algebras
Let K be a commutative field. The author shows that every radical K- algebra (i.e. coinciding with its Jacobson radical) can be embedded in a radical K-algebra A with the property that A contains no nonzero ideal which is nil or residually nilpotent or satisfies a polynomial identity. The construction is analogous to that used by \textit{L. A. Bokut'} [Sib. Mat. Zh. 4, 500-518 (1963; Zbl 0136.022)] to prove that for any \(a\neq 0\) and b, two elements u, v can be adjoined to satisfy \(b=uav\). Secondly, define an algebra to be m-nilpotent if each subalgebra generated by m elements is nilpotent. Then the author shows that every locally nilpotent (m-nilpotent) K-algebra can be embedded in a locally nilpotent (resp. m- nilpotent) K-algebra A containing no nonzero ideal which is nil of bounded index or satisfies a polynomial identity. Using methods of Golod, he deduces that every nil K-algebra can be embedded in a nil K-algebra whose locally nilpotent ideals are totally ordered by inclusion.
scientific article; zbMATH DE number 3912552
scientific article