Prime Submodules

[摘要] Let R be a ring. A proper submodule K of an 72-module 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 R-module s.t.r.f.. In [25] McCasland and Moore proved that a commutative ring R s.t.r.f. provided that every free 72-module 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 non-finitely generated submodules of free modules are studied. Let M1, M2 be R-modules 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 R-module R s.t.r.f. but the R-module 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 R-modules 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 R-modules, then the R-module 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 thisresult 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 R-modules which do not satisfy the ascending chaincondition 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 R-module 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.

[发布机构] University:University of Glasgow

[关键词] Mathematics