Commutative Local Rings of bounded module type

Let R be a local ring of bounded module type. It is shown that R is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is an almost maximal valuation domain. We deduce from this that R is almost maximal if one of the following conditions is satisfied: R is a Q-algebra of Krull dimension ≤ 1 or the maximal ideal of R is the union of all non-maximal prime ideals. In this paper, R is an associative and commutative ring with identity. We will say that R has bounded module type if, for some positive integer n, every finitely generatedR-module is a direct sum of submodules generated by at most n elements. The problem of investigating commutative rings of bounded module type has been studied by R.B. Warfield [9], R. Wiegand [10], B. Midgarden and S. Wiegand [6], P. Zanardo [11], P. Vámos [8] and the author [1]. In [9], R.B. Warfield proved that every local ring of bounded module type is a valuation ring. By theorems due to D.T. Gill [4] and J.P. Lafon [5], a valuation ring is almost maximal if and only if every finitely generated module is a direct sum of cyclics. So, the following question can be proposed : Is a local ring R of bounded module type if and only if R is an almost maximal valuation ring ? Positive answers are given by P. Zanardo in [11] for the class of totally branched and discrete valuation domains, by P. Vámos in [8] for Q-algebra valuation domains and in [1] by the author for valuation rings with a finitely generated maximal ideal. In this paper, we prove that if R is a valuation ring of bounded module type, then R/I is complete in its ideal topology for every nonzero and nonarchimedean ideal I, and RJ is almost maximal for every nonmaximal prime ideal J . To obtain these results, as in [1], we adapt to the nondomain case Zanardo’s methods used in [11]. Moreover, we extend results obtained by P. Vámos in [8] : every valuation ring R of bounded module type is almost henselian, and if R is a Q-algebra with Krull dimension greater than one, then R is almost maximal. Finally we obtain also a positive answer for valuation rings such that the maximal ideal is the union of all nonmaximal prime ideals. For definitions and general facts about valuation rings and their modules we refer to the book by Fuchs and Salce [2]. The symbol A ⊂ B denotes that A is a subset of B, possibly A = B. We recall some definitions and results which will be used in the sequel. An R-module M is called fp-injective if and only if Ext(F,M) = 0 for every finitely presented R-module F . A self fp-injective ring R is fp-injective as R-module. For an R-module M and an ideal I of R, we set M [I] = {x ∈ M | I ⊂ (0 : x)}. If M is fp-injective then rM = M [(0 : r)] for every r ∈ R. It is obvious that rM ⊂ M [(0 : r)]. Now let x ∈ M [(0 : r)]. It 1