Narrow Results

A base of the free alternative nil-superalgebra of index 3 on one odd generator is constructed. In particular, its index of solvability is computed. We consider also the corresponding Grassmann algebra and show that the well-known Dorofeev's example of solvable non-nilpotent alternative algebra is its homomorphic image.

For $n\ge 2$ and for a ring $R,$ the notation $P_n(R)$ means that $r^n-r$ is nilpotent for all $r\in R$. In this paper, rings $R$ for which $P_n(R)$ holds are completely characterized for any integers $n\ge 2$. This answers a question which was raised in [T. Kosan, Y. Zhou, T. Yildirim, Rings with $x^n-x$ nilpotent, J. Algebra Appl.19(4) (2020) 2050065].

In this paper we study flexible algebras (possibly infinite-dimensional) satisfying the polynomial identity x(yz) = y(zx). We prove that in these algebras, products of five elements are associative and commutative. As a consequence of this, we get that when such an algebra is a nil-algebra of bounded nil-index, it is nilpotent. Furthermore, we obtain optimal bounds for the index of nilpotency. Another consequence that we get is that these algebras are associative when they are semiprime.