Characterizing a class of simply presented modules by relation arrays

Simply presented modules and Warfield modules are described in a class of m ixed modules H with the property that the torsion submodule is a direct sum of cyclics and the quotient module the torsion submodule is divisible of a rbitrary rank. Analogous to a result of Warfield it is shown that the mixed modules of torsion-free rank one are in some sense the building blocks of such modules. The results extend our previous work describing this class by relation arrays which are a natural outgrowth of their basic generating sy stems. Moreover, an intimate connection is shown between relation arrays an d the indicators of modules. Furthermore, we prove two realizability result s one of which is analogous to a theorem of Megibben for mixed modules of t orsion-free rank one. One gives necessary and sufficient conditions on when a relation array of a module can realize an indicator of finite-type while the other shows that an admissable indicator can be realized by a simply p resented module of torsion-free rank one.