Finitely Presented Modules over Semihereditary Rings

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring. It is well-known for a long time that every Dedekind domain satisfies the stacked bases property (or the Simultaneous Basis Property). See [8]. The definitions will be given below. More recently this result was extended to every Prüfer domain of finite character [10] or of Krull dimension one [2] and more generally to each almost local-global Prüfer domain [2]. The aim of this paper is to show that every almost local-global semihereditary ring, and in particular every hereditary ring, has the stacked bases property too. This extension is possible principally because every semihereditary ring has the following properties: • every finitely presented module is the direct sum of its torsion submodule with a projective module, • the annihilator of each finitely generated ideal is generated by an idempo-tent, • and each faithful finitely generated ideal contains a nonzerodivisor. So, the only difference with the domain case is that we have to manage nontrivial idempotents. Except for this difference, we do as Section 4 of Chapter V of [5]. All rings in this paper are unitary and commutative. We say that a ring R is of finite (resp. countable) character if every non-zero-divisor is contained in finitely (resp. countably) many maximal ideals. We say that R is local-global if each polynomial over R in finitely many indeterminates which admits unit values locally, admits unit values. A ring R is said to be almost local-global if for every faithful …