# Leavitt path algebras with bounded index of nilpotence

Source: [Leavitt path algebras with bounded index of nilpotence](<https://www.sota2.com/research/paper/68fa552b1e209131ce991974>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T13:24:40.416Z

Authors: K. M. Rangaswamy, Ashish K. Srivastava

Published: 2018-08-30T01:50:08.000Z

## Abstract

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra $L\_{K}(E)$ has index of nilpotence at most $n$ if and only if no cycle in the graph $E$ has an exit and there is a fixed positive integer $n$ such that the number of distinct paths that end at any given vertex $v$ (including $v$, but not including the entire cycle $c$ in case $v$ lies on $c$) is less than or equal to $n$. 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 $\\mathbb{Z}$-graded $\\Sigma$-$V$ rings. As an application of our results, we answer an open question raised in \\cite{JST} whether an exchange $\\Sigma$-$V$ ring has bounded index of nilpotence.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1808.10756>)

[PDF](<https://arxiv.org/pdf/1808.10756.pdf>)