On the combinatorics of extensions of preinjective Kronecker modules

The Extension Monoid Product of Preinjective Kronecker Modules

We explore the combinatorial properties of the extension monoid product of preinjective Kronecker modules. We state a theorem which characterizes the extension monoid product of preinjective (and dually preprojective) Kronecker modules in the most general case, over an arbitrary base field. As corollaries, we give another proof of an interesting theorem from [7] and restate the main theorem fro...

Subrepresentations of Kronecker Representations

Translated into the language of representations of quivers, a challenge in matrix pencil theory is to find sufficient and necessary conditions for a Kronecker representation to be a subfactor of another Kronecker representation in terms of their Kronecker invariants. The problem is reduced to a numerical criterion for a Kronecker representation to be a subrepresentation of another Kronecker rep...

Extensions of Baer and quasi-Baer modules

On the Endofiniteness of a Key Module over Pure Semisimple Rings

Let R be a left pure semisimple ring such that there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, and let W be the left key R-module; i.e., W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R-modules. We show that if the module W is endofinite, then R is a ring...

Preprojective Modules and Auslander-Reiten Components

In [2], Auslander and Smalø introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. In [4], Coelho studied Auslander-Reiten components containing only hereditarily preproj...