Leavitt path algebras with bounded index of nilpotence
In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra [Formula: see text] has index of nilpotence at most [Formula: see text] if and only if no cycle in the graph [Formula: see text] has an exit and there is a fixed positive integer [Formula: see text] such that the number of distinct paths that end at any given vertex [Formula: see text] (including [Formula: see text], but not including the entire cycle [Formula: see text] in case [Formula: see text] lies on [Formula: see text]) is less than or equal to [Formula: see text]. Interestingly, the Leavitt path algebras having bounded index of nilpotence turn out to be precisely those that satisfy a polynomial identity. Furthermore, Leavitt path algebras with bounded index of nilpotence are shown to be directly-finite and to be [Formula: see text]-graded [Formula: see text]–[Formula: see text] rings. As an application of our results, we answer an open question raised in [S. K. Jain, A. K. Srivastava and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford Mathematical Monographs (Oxford University Press, 2012)] whether an exchange [Formula: see text]–[Formula: see text] ring has bounded index of nilpotence.