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 (we say such a graph satisfies Condition (AR)). 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 satisfying Condition (AR) the understanding of injective envelopes of simple modules over $L_K(E)$ achieved previously for the simple modules over the Toeplitz algebra.