Modules with RD-composition series over a commutative ring

If R is a commutative ring, then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring, the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these numbers are equal if and only if M is a direct sum of cyclic modules. We also give an example of an artinian module over a noetherian domain, which has an RDcomposition series with uniserial factors. Finally we prove that every pure-injective R-module is RD-injective if and only if R is an arithmetic ring. In this paper, for a commutative ring R, we study the following properties: (1) Every finitely generated R-module M has a finite chain of RD-submodules with cyclic factors. (2) Every finitely generated R-module M has a finite chain of RD-submodules with indecomposable cyclic factors. (3) R satisfies (2) and any two chains of RD-submodules of M, with indecomposable cyclic factors, are isomorphic. In [1], L.Salce and P.Zanardo proved that every valuation ring satisfies (3), and in [2] C.Naudé showed that each h-local and Bézout domain also satisfies (3). In section 1, we give definitions and some preliminary results. It is proved that every ring that satisfies (1), is a Kaplansky ring(or an elementary divisor ring), but we don’t know if the converse holds. In section 2, we state that a ring R satisfies (3) if and only if R is a Bézout ring and a CF-ring([3]). We show that R satisfies (2), when R is a semilocal arithmetic ring or an h-semilocal Bézout domain, and for every finitely generated R-module M, we prove that any chain of RD-submodules of M has the same length which is equal to the number of terms of a finite sequence of prime ideals, associated to M. However, we give two examples of semilocal arithmetic rings that don’t satisfy (3). In section 3, when R is a ring that verifies (3), the length l(M) of each chain of RD-submodules, with indecomposable cyclic factors, is compared with g(M), the Goldie dimension of M. We show that g(M) ≤ l(M) and that M contains a direct sum of g(M) nonzero indecomposable cyclic submodules which is an essential RDsubmodule of M. Hence it follows that M is a direct sum of cyclic submodules if and only if g(M) = l(M). These results were proved in [1], when R is a valuation ring and in [2], when R is an h-local Bézout domain. Finally, in section 4, we give an example of an artinian module, over a noetherian domain, that has a finite chain of RD-submodules, with uniserial factors. We also give explicit examples of pure-injective modules that fail to be RD-injective, over