Indecomposable injective modules of finite Malcev rank over local commutative rings

It is proven that each indecomposable injective module over a valuation domain R is polyserial if and only if each maximal immediate extension R̂ of R is of finite rank over the completion R̃ of R in the R-topology. In this case, for each indecomposable injective module E, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over local Noetherian rings of Krull dimension one has finite Malcev rank. The preservation of the finiteness of Goldie dimension by localization is investigated too. Introduction and preliminaries In this paper all rings are associative and commutative with unity and all modules are unital. First we give some definitions. Definition 0.1. An R-module M is said to be uniserial if its set of submodules is totally ordered by inclusion and R is a chain ring if it is uniserial as R-module. A chain domain is a valuation domain. In the sequel, if R is a chain ring, we denote by P its maximal ideal, N its nilradical, Z its set of zero-divisors (Z is a prime ideal) and we put Q = RZ . Recall that a chain ring R is said to be Archimedean if P is the sole non-zero prime ideal. A module M is said to be finitely cogenerated if its injective hull is a finite direct sum of injective hulls of simple modules. The f.c. topology on a module M is the linear topology defined by taking as a basis of neighbourhoods of zero all submodules G for which M/G is finitely cogenerated (see [17]). This topology is always Hausdorff. We denote by M̃ the completion of M in its f.c. topology. When R is a chain ring which is not a finitely cogenerated R-module, the f.c. topology on R coincides with the R-topology which is defined by taking as a basis of neighbourhoods of zero all non-zero principal ideals. A chain ring R is said to be (almost) maximal if R/A is complete in its f.c. topology for any (non-zero) proper ideal A. In 1959, Matlis proved that a valuation domain R is almost maximal if and only if Q/R is injective, and in this case, for each proper ideal A of R, E(R/A) ∼= Q/A, see [15, Theorem 4]. Since Q is clearly uniserial and Q/R ∼= Q/rR for each non-zero element r ∈ P , we can also say that R is almost maximal if and only if E(R/rR) 2010 Mathematics Subject Classification. 13F30, 13C11, 13E05.