2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146357Let G be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let g be its Lie algebra. First we extend a well-known result about the Picard group of a semisimple group to reductive groups. Then we prove that, if the derived group is simply connected and g satisfies a mild condition, the algebra K[G]^g of regular functions on G that are invariant under the action of g derived from the conjugation action, is a unique factorisation domain.Commutative AlgebraRings and Algebrastext