Quasi-projective modules with the finite exchange property

Stable modules and a theorem of Camillo and Yu

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.

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with M = A + B and M / A ≅ M / B . The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is DSF. Every DSF-module M is Dedekind finite, and if M satisfies the finite-exchange property, then M satisfies the substitution property and its endomorphism ring has stable range 1. Moreover, a DSF-module M has the finite exchange property iff M is clean, iff M has the full exchange property. Finally, maximal submodules of DSF-modules are shown to be fully-invariant, in particular a ring R is DSF as a right R-module iff R is right quasi-duo (i.e. every maximal right ideal is two-sided). The later result provides a new perspective on the symmetry question of quasi-duo rings.

Square-free modules with the exchange property