Straight and Strongly Straight Primary Modules over Principal Ideal Domains

Straight and strongly straight primary abelian groups were introduced by the first author and K. Honda in [2]. It is commonly believed that most results in abelian group theory carry over to modules over principal ideal domains with only minor adjustments. It turns out that the concept of straightness and strong straightness behave differently according to the principal ideal domains chosen. This phenomenon justifies the study of these concepts in the more general setting of primary modules over arbitrary principal ideal domains. We obtain the following characterization: A primary module is strongly straight if and only if every isometry between its socle and that of a straight module extends to an isomorphism. As a consequence of this characterization the class of strongly straight primary modules is seen to contain all torsion complete primary modules, all direct sums of cyclic primary modules and all divisible primary modules. We show also that every subsocle of a strongly straight primary module M supports a pure submodule K which is straight and M/K is straight. In particular strongly straight primary modules belong to the elusive class of pure complete modules. Over the polynomial ring K[t] where K is a field, straight t-primary modules are strongly straight, however over ZZ the reduced Prufer group is straight but not strongly straight.