2018/08/30 by K. M. Rangaswamy, Rangaswamy, K. M., Ashish K. Srivastava +1
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.1808.10756
arXiv admin note: substantial text overlap with arXiv:1705.09217, arXiv:1611.07858, To appear in J. Algebra and Appl
arxiv created 2018/09/18 · arxiv updated 2018/09/19
In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra LK(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 ℤ-graded Σ-V rings. As an application of our results, we answer an open question raised in \citeJST whether an exchange Σ-V ring has bounded index of nilpotence.