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 LK(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 LK(E) achieved previously for the simple modules over the Toeplitz algebra.