Square-free modules with the exchange property

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.

Stable modules and a theorem of Camillo and Yu

It is proven that every purely infinite simple ring is an exchange ring. This result is applied to determine those Leavitt algebras that are exchange rings.

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.

Quasi-projective modules with the finite exchange property

Let Mk be a right k-module with endomorphism ring E=End(Mk). We prove that if E is an Abelian exchange ring, then Mk has the full exchange property. We also give an extension of this result in the case E is regular.