Let \(D\) be a domain with quotient field \(K,\) throughout. An ideal \(I\) of \(D\) is called locally principal if \(ID_{P}\) is principal for each prime ideal \(P\) of \(D.\) A domain \(D\) is called an LPI domain if every nonzero locally principal ideal of \(D\) is invertible. LPI domains were studied by Anderson and the reviewer in [\textit{D. D. Anderson} and \textit{M. Zafrullah}, Commun. Algebra 39, No. 3, 933--941 (2011; Zbl 1225.13001)]. Next, a flat \(D\) -module \(M\) is \textit{faithfully flat if }\(M\otimes _{D}N=0\) implies \(N=0\) for any \(D\)-module \(N.\) Also it is well-known that a nonzero ideal \(I\) of \(D\) is faithfully flat if and only if \(I\) is locally principal, \textit{S. Glaz} and \textit{W. V. Vasconcelos} [Manuscr. Math. 22, 325--341 (1977; Zbl 0367.13002)]. Thus an LPI domain is precisely a domain in which every faithfully flat ideal is invertible. Noetherian domains are clearly LPI domains. Yet there are domains that may not have this property. The proof of Proposition 7 of the reviewer's [J. Pure Appl. Algebra 214, No. 5, 654--657 (2010; Zbl 1188.13010)] shows, there do exist domains \(D\) in which faithfully flat ideals may not be invertible. Other related concepts are rings whose flat ideals are finitely generated. These rings were recently studied under the title of ff-rings in [\textit{S. El Baghdadi}, \textit{A. Jhilal} and \textit{N. Mahdou}, J. Pure Appl. Algebra 216, No. 1, 71--76 (2012; Zbl 1239.13002)], and were studied as rings with property \(\mathcal{P}\) in [Commun. Algebra 3, 531--543 (1975; Zbl 0315.13010)] by \textit{J. D. Sally} and \textit{W. V. Vasconcelos}. Indeed, as a nonzero finitely generated flat ideal is invertible in an integral domain, an ff-domain \(D\) is an LPI domain. Yet an LPI domain is not always an ff-ring, as in a rank one non-discrete valuation ring the maximal ideal is flat but not finitely generated. Now, Sally and Vasconcelos [Zbl 0315.13010] showed that if \(D\) is an ff-domain then so is \(D[X]\), and El-Baghdadi et al. [Zbl 1239.13002] prove the converse. For LPI domains, Anderson and the reviewer [Zbl 1225.13001] proved, using multiplicative ideal theory techniques, that if \(D[X]\) is an LPI domain then so is \(D\) and if \(D\) were integrally closed then the converse would be true too, leaving out the proof of the converse as an open problem, for \(D\) not integrally closed. The authors of the paper under review show, using modifications of techniques of Sally and Vasconcelos [Zbl 0315.13010] that if \(D\) is an LPI domain then so is \(D[X]\) and extend the study to locally finitely generated and locally free modules ending with the following result. The following statements are equivalent for a domain \(D\). (1) \(D\) is an LPI domain. (2) Every locally finitely generated and locally free submodule of \(D^{n}\) of rank \(n\) over \(D\) is finitely generated.