Narrow Results

We study a construction introduced by Drápal, giving rise to commutative A-loops of order kn where k and n are odd numbers. We show which combinations of k and n are possible if the construction is based on a field or on a cyclic group. We conclude that if p and q are odd primes, there exists a non-associative commutative A-loop of order pq if and only if p divides q² - 1 and such a loop is most probably unique.

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].