Throughout, all rings are commutative with non-zero identity and all modules are unitary. Let R be a ring and let M be an R-module. A proper submodule N of M is called p-prime (resp. p-primary) if rm ∈ N for r ∈ R and m ∈ M implies that either m ∈ M or r ∈ p = (N : M) (resp. m ∈ M or r ∈ p = √(N : M). The radical of N in M, denoted by rad_M(N), is defined to be intersection of all prime submodules of M containing M, or rad_M(N) = M in case no prime submodule of M contains N. The envelope of N in M is the set of elements rm of M such that r ∈ R,m ∈ M and r^km ∈ N for some positive integer n. In general, E_M(N) is not a submodule of M, for a given submodule N. We denote by < E_M(N) > the submodule of M generated by the set E_M(N).

In this paper, we consider the problem of finding a generating set of the radical and envelope of a submodule N of R-module M from given generating set the submodule when R = k[x₁, . . . , x_n] is the polynomial ring over a field k and M is free module R^m for some positive integer m. Although a closely related problem that finding primary decomposition of a submodule in above setting has been extensively studied, it seems to be there is no method eveloped for finding a generating set of radical or envelope of a submodule. If q is a primary ideal of a ring, then it is well-known that rad(q) is a prime ideal. In module case, however, Q a primary submodule of M does not necessarily imply that rad_M(Q) is a prime submodule.
Moreover, it is not always true for submodules N and L of M that rad_M(N \ L) = rad_M(N) \ rad_M(L). Hence knowing a primary decomposition of a submodule N does not automatically give a generating set for radical of that submodule.