Characterizing a class of simply presented modules by relation arrays

Abstract

Simply presented modules and Warfield modules are described in a class of mixed modules h with the property that the torsion submodule is a direct sum of cyclics and the quotient modulo the torsion submodule is divisible of arbitrary 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 systems. Moreover, an intimate connection is shown between relation arrays and the indicators of modules. Furthermore, we prove two realizability results one of which is analogous to a theorem of Megibben for mixed modules of torsion-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 presented module of torsion-free rank one. 0. Introduction. This paper characterizes the simply presented modules in a class of mixed modules h with the property that the torsion submodule is a direct sum of cyclics and the quotient modulo the torsion submodule is divisible of arbitrary rank. This class was described by generators and relations and the concept of a relation array and its properties studied in a series of papers [4,6,7,8]. In [6] we gave a splitting criterion in terms of the relation array which was extended in [7] to a larger class where the quotient p-rank is finite. In this paper we show the connection between the calculation of indicators done in [4] and the relation arrays. We then use this to describe the simply presented and Warfield modules in the class h.

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

“…[6,7,8,9]. As in [6] we denote by h 1 the subclass of h consisting of the modules of torsion-free rank one.…”