Nil modules and the envelope of a submodule

Let R be a commutative unital ring and N be a submodule of an R-module M. The submodule EM(N) generated by the envelope EM(N) of N is instrumental in studying rings and modules that satisfy the radical formula. We show that: 1) the semiprime radical is an invariant on all the submodules which are respectively generated by envelopes in the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, EM(0) is an idempotent radical and it induces a torsion theory whose torsion class consists of all nil R-modules and the torsionfree class consists of all reduced R-modules; 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module; and lastly, 4) we construct a sheaf of nil R-modules on Spec(R).