2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59124Simple necessary and sufficient conditions for a $n$-tuple of noncommutative polynomials to be a cyclic gradient are given and similarly for a noncommutative polynomial to have a vanishing cyclic gradient. Connections with free probability are explained.5 pagesRings and AlgebrasOperator Algebras16W99; 46L54, 16S10, 12H99text