Dual Perspectively Decomposable Modules

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.