Largest ideals in Leavitt path algebras (Q2178620)

The paper describes ideals of Leavitt path algebras of arbitrary not necessarily connected digraphs generated by certain subsets of vertices. Namely, the authors consider set \(P_l\), \(P_c\), \(P_{ec}\), \(P_{b^{\infty}}\) of vertices, respectively, which consists, in order, of all line points, vertices in cycles without exits or of extreme cycles and vertices whose trees have infinitely many bifurcations or at least one infinite emitter. Further subsets of vertices are also involved. The main goal of the article is to show that ideals generated by these subsets are largest ideals in certain sense. In particular, the ideal \(I(P_l)\) generated by line points which is already known as the socle, is the largest locally Artinian ideal, while the ideal \(I(P_c)\) generated by \(P_c\), is the largest locally Noetherian ideal without minimal idempotent. The paper characterizes also both the largest purely infinite ideal and the largest exchange ideal of Leavitt path algebras. Further properties of these ideals are also investigated. The paper contains several nice results but is difficult to read.