Valuation, discrete valuation and Dedekind modules (Q2920177)

The authors define \textit{valuation modules} and \textit{discrete valuation modules} over an integral domain,which are generalisations of valuation rings and discrete valuation rings, respectively. Using many basic results on valuation and discrete valuation modules, which are derived in the first two parts of the paper, the authors characterise Dedekind modules via discrete valuation modules. In particular, a Noetherian faithful multiplication \(R\)-module such that every nonzero prime submodule is maximal, where \(R\) is a domain, is a Dedekind module if and only if the localisation of \(M\) at \(\mathfrak{p}\) is a discrete valuation module for all nonzero primes \(\mathfrak{p}\) in the Spec of \(R\). Similar results are also presented.