Leavitt path algebras with bounded index of nilpotence

Abstract

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.