Injectives over Leavitt path algebras of graphs with disjoint cycles

Let K be any field, and let E be a finite graph with the property that every vertex in E is the base of at most one cycle (i.e., a graph with disjoint cycles). We explicitly construct the injective envelope of each simple left module over the Leavitt path algebra L_K(E). The main idea girding our construction is that of a “formal power series” extension of modules, thereby developing for all graphs with disjoint cycles the understanding of injective envelopes of simple modules over L_K(E) achieved previously for the simple modules over the Toeplitz algebra.