Let R be a ring. A proper submodule K of an 72module M is called prime if whenever r ∈ R, m ∈ M and rRm ⊆ K then m ∈ K or rM ⊆ K. It is clear that prime submodules generalize the usual notion of prime ideals. The radical of a submodule N of M, denoted by radM(N) is defined to be the intersection of all prime submodules of M containing N. Now let R be a commutative ring. Let 7 be an ideal of R, As is well known, the radical of 7, defined as the intersection of all prime ideals containing 7, has the characterization √I = {r ∈ R : rn ?∈ I, for some n ∈ Z+}. A natural question arises, whether there is a somewhat similar characterization for the radical of a submodule, in particular, a characterization in which the knowledge of prime submodules(indeed even prime ideals) is not necessary. Under certain conditions such a characterization is provided by the concept of the envelope of a submodule. The envelope of N, EM(N), is the collection of all m ∈ M for which there exist r ∈ R, a ∈ M such that m = ra and rna ∈ N for some positive integer n. Always Em (N) ⊆ radM(N). We say that M satisfies the radical formula (M s.t.r.f.) if for every submodule N of M radM(N) =, the submodule of M generated by EM(N). A ring R s.t.r.f. provided that every Rmodule s.t.r.f.. In [25] McCasland and Moore proved that a commutative ring R s.t.r.f. provided that every free 72module F s.t.r.f.. Accordingly, in chapter 2, prime submodules of free modules over commutative domains are investigated. A fundamental question in the study of prime submodules is how to describe radM(N) for a given submodule N of a module M. In the first section of chapter 3, radF(N) is described where N is is a finitely generated submodule of the free module F. In the second section the radicals of some nonfinitely generated submodules of free modules are studied. Let M1, M2 be Rmodules such that M1 ⊕ M2 s.t.r.f.. Then M1 and M2 both s.t.r.f.. The converse is not true in general. For example, if R is a Noetherian domain which is not Dedekind then the Rmodule R s.t.r.f. but the Rmodule R ⊕ R does not. But it is true in some cases and this is considered in the first section of chapter 4. For example, if R is a commutative ring and M1, M2 are Rmodules such that M1 s.t.r.f. and M2 is semisimple, then M1 ⊕ M2 s.t.r.f.. Also if A is a finite direct sum of cyclic Artinian Rmodules, then the Rmodule R ⊕ A s.t.r.f.. The aim of the second section is to describe EF(N) in a nice way, where N is a finitely generated submodule of a free module F of finite rank. In [9] Gordon and Robson proved that any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals, but this result does not hold for modules in general. In particular, if R is the first Weyl algebra over a field of characteristic 0 then there are Artinian Rmodules which do not satisfy the ascending chain condition on semiprime submodules. The aim of chapter 5 is to investigate when Gordon and Robson's result holds for modules. It is proved that if R is a ring which satisfies a polynomial identity then any Rmodule with Krull dimension satisfies the ascending chain condition on prime submodules, and, if R is left Noetherian, also the ascending chain condition on semiprime submodules.
