Modules with local Pierce stalks (Q1183306)

For the purpose of this paper, an \(R\)-module \(A\) is called local if it contains a (unique) maximal submodule which contains every proper submodule of \(A\). Let \(B(R)\) denote the Boolean ring of central idempotents of a ring \(R\) and let \(X\) denote the spectrum of maximal ideals of \(B(R)\). If \(x\in X\), set \(S=B(R)\backslash x\). The Pierce stalk of a left \(R\)-module \(A\) at \(x\) is defined as \(A_ x=s^{-1}A\) and is isomorphic to \(A/xA\). \(A\) is said to have local Pierce stalks if \(A_ x\) is a local \(R\)-module for every \(x\in X\). It is shown that \(A\) has local Pierce stalks if and only if there is an \(a\in A\) such that every element of \(A\) is of the form \(c+ea\), where \(c\) is a generator of \(A\) and \(e\) is a central idempotent of \(R\). This is a generalization of a theorem by Burgess and Stephenson on the existence of local Pierce stalks for a ring. An \(R\)-module \(A\) is called suitable if for each pair of comaximal submodules \(B+C=A\), there exists a direct summand \(B'\) of \(A\) contained in \(B\) and comaximal to \(A\). It is proved that \(_ RA\) has local Pierce stalks if and only if \(_ RA\) is suitable and every idempotent of the endomorphism ring \(E\) of \(A\) is in the image of \(B(R)\). Suitability is shown to be equivalent to having the finite exchange property for quasi-projective modules, leading to a theorem on the existence of local Pierce stalks for finitely generated quasi- projective modules in terms of the exchange property. This again is a generalization of work of others, from projectives to quasi-projectives. An example is given showing that the theorem does not hold without the hypothesis of quasi-projectivity. The paper concludes with a short proof that a finitely generated quasi-projective module is local if and only if its endomorphism ring is local.