A module Formula: see text is called dual perspectively indecomposable if, Formula: see text does not contain proper perspectively related submodules Formula: see text and Formula: see text with Formula: see text, where two submodules Formula: see text and Formula: see text of Formula: see text are called perspectively related, and denoted by Formula: see text, if Formula: see text, for a submodule Formula: see text. Every indecomposable module is dual perspectively indecomposable, but the converse is not true. Moreover, Formula: see text is called dual perspectively decomposable (dual PD-module) if, Formula: see text for every pair of proper submodules Formula: see text and Formula: see text of Formula: see text with Formula: see text and Formula: see text. Examples are provided to show that the class of dual Formula: see text-modules lies strictly between the classes of summand-dual-square-free and Formula: see text-modules. We will show that every dual Formula: see text-module is a finite direct sum of dual perspectively indecomposable submodules. As an application, we prove that if Formula: see text is a dual Formula: see text-module with the finite exchange, then Formula: see text is clean and has the full exchange. This is a partial answer to Crawley-Jónsson’s open question that asks whether the finite exchange property of a module implies the full exchange property.
Das et al. (Fri,) studied this question.