Injective envelopes for Chen simples arising from non-exclusive cycles

Describe the injective envelope of the Chen simple left module V^E_c when c is a non-exclusive cycle, in particular for the L_K(R_2)-module V^{R_2}_c associated with either of the two loops in the graph R_2 having one vertex and two distinct loops.

Background

The paper constructs explicit injective envelopes for Chen simple modules VE_c and, more generally, VE_{p(x),c}, when c is an exclusive cycle—that is, no vertex on c is the base of a cycle other than c itself. The construction uses extended Prüfer modules realized as source-finite formal series and applies to arbitrary directed graphs.

The authors explicitly identify the corresponding problem for non-exclusive cycles as unresolved. They single out the graph R_2 with one vertex and two distinct loops; because each loop shares its base vertex with the other loop, neither loop is exclusive. Determining the injective envelope of the associated Chen simple module would extend the paper’s results beyond the exclusive-cycle case.

References

That being said, we readily admit that there is still significant work to be done in this regard. In particular, we have not yet been able to describe the injective envelope of a Chen simple module of the form $VE_c$ where $c$ is a non-exclusive cycle. Specifically, we have not yet been able to describe the injective envelope of the $L_K(R_2)$-module $V{R_2}_{c}$, where

R_2: =\quad\xymatrix{\bullet\ar@(dl,ul)\ar@(ur,dr)} and $c$ is one of the two (non-exclusive) loops in $R_2$.

— The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles  (2609.00952 - Abrams et al., 1 Sep 2026) in Introduction, concluding paragraph before Section 2