An R-module M is called strongly duo if Tr(N, M) = N for every submodule N of M. Several equivalent conditions to being strongly duo are given. A submodule N of M is called totally fully invariant if f(K) is contained in N for all and . M R is strongly duo if and only if every submodule of M is totally fully invariant. If M R is strongly duo and reduced, then EndR(M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M R is strongly duo if and only if either or M R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every piojective strongly duo module is a multiplication module. A ring R is called strongly duo if R R is strongly duo. Strongly regular rings are precisely reduced strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring. An R-module M is called weakly endosimple if it has no nontrivial totally fully invariant submodule. A ring R is called right weakly endosimple if R R is weakly endosimple. Commutative domains are precisely commutative weakly endosimple rings.