Let \(R\) be a discrete valuation domain and \(\mathcal H\) the class of \(R\)-modules \(G\) with the property that the torsion submodule \({\mathbf t}G\) of \(G\) is a direct sum of cyclic submodules and the quotient module \(G/{\mathbf t}G\) is divisible of any rank. The current paper gives necessary and sufficient conditions for when certain types of modules in \(\mathcal H\) are Warfield modules. For each major result an example is given which illustrates the tightness of the fit of the conditions. The proofs rely on describing modules by generators and relations, their corresponding relation arrays, and earlier theory established by these same authors.
Some of the main results include a more generalized version of (1) If \(G\) is a reduced \(R\)-module, then it is Warfield if and only if its first Ulm submodule is free. And (2) If an \(R\)-module \(G\) in the class \(\mathcal H\) is either strictly reduced or has a torsion Ulm factor, then the following are equivalent: (a) \(G\) is Warfield, (b) \(G\) is simply presented, (c) \(G\) is a direct sum of modules of torsion-free rank 1.