Local rings of bounded module type are almost maximal valuation rings

It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring. We say that an associative commutative unitary ring R is of bounded module type if there exists a positive integer n such that every finitely generated R-module is a direct sum of submodules generated by at most n elements. For instance, every Dedekind domain has bounded module type with bound n = 2. Warfield [9, Theorem 2] proved that every commutative local ring of bounded module type is a valuation ring. Moreover, Gill [5] and Lafon [7] showed independently that a valuation ring R is almost maximal if and only if every finitely generated R-module is a direct sum of cyclic modules. So every almost maximal valuation ring has bounded module type and Vámos proposed the following conjecture, in the Udine Conference on Abelian Groups and Modules, held in 1984: “A local ring of bounded module type is an almost maximal valuation ring.” P. Zanardo [10] and P. Vámos [8] first investigated this conjecture and proved it respectively for strongly discrete valuation domains and Q-algebra valuation domains. More recently the author proved the following theorem (see [1, Corollary 9 and Theorem 10]). Theorem 1. Let R be a local ring of bounded module type. Suppose that R satisfies one of the following conditions: (1) There exists a nonmaximal prime ideal J such that R/J is almost maximal. (2) The maximal ideal of R is the union of all nonmaximal prime ideals. Then R is an almost maximal valuation ring. From this theorem we deduce that it is enough to show the following proposition to prove Vámos’ conjecture : see [1, remark 1.1]. Recall that a valuation ring is archimedean if the maximal ideal is the only non-zero prime ideal. Proposition 2. Let R be an archimedean valuation ring for which there exists a positive integer n such that every finitely generated uniform module is generated by at most n elements. Then R is almost maximal. In the sequel we use the same terminology and the same notations as in [1] except for the symbol A ⊂ B which means that A is a proper subset of B. We also use the terminology of [2]. Some results of [2] and the following lemmas will be useful to show proposition 2. An R-module E is said to be fp-injective(or absolutely pure) if Ext1R(F,E) = 0, for every finitely presented R-module F. A ring R is called self fp-injective if it is fp-injective as R-module.