A module is called semi-endosimple if it has no proper fully invariant essential submodules.Basic properties of semi-endosimple modules are explored, and rings with all finitely generated modules semi-endosimple are characterized. It is proved that a hereditary left Noetherian ring has all finitely generated modules semi-endosimple if and only if it is a finite direct sum of simple Noetherian V-rings. We carry out a study of modules atisfying the property that every module in is a Kasch module. Such modules are called fully Kasch. When R is Morita equivalent to a right duo ring, is fully Kasch if and only if is a left perfect ring for any non-zero . These considerations tackle a question raised by Albu and Wisbauer.