2. Two-Sided Ideals

In this chapter we investigate the ideal structure of Leavitt path algebras. We start by describing the natural \(\mathbb{Z}\)-grading on L _K (E). We then present the Reduction Theorem; this result describes how elements of L _K (E) may be transformed in some specified way to either a vertex or a cycle without exits. Numerous consequences are discussed, including the Uniqueness Theorems. We then establish in the Structure Theorem for Graded Ideals a precise relationship between graded ideals and explicit sets of idempotents (arising from hereditary and saturated subsets of vertices, together with breaking vertices). With this description of the graded ideals having been achieved, we focus in the remainder of the chapter on the structure of all ideals. We achieve in the Structure Theorem for Ideals an explicit description of the entire ideal structure of L _K (E) (including both the graded and non-graded ideals) for an arbitrary graph E and field K. This result utilizes the Structure Theorem for Graded Ideals together with the analysis of the ideal generated by vertices which lie on cycles having no exits. A number of ring-theoretic results follow almost immediately from the Structure Theorem for Ideals, including the Simplicity Theorem. Along the way, we describe the socle of a Leavitt path algebra, and we achieve a description of the finite dimensional Leavitt path algebras.