arXiv subjects

K. M. Rangaswamy

Publications and source records attributed to K. M. Rangaswamy.

4 recordsLinked to original sources

arXiv2021

Variations of primeness and Factorization of ideals in Leavitt Path Algebras

In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under which a proper ideal of a Leavitt path algebra $L$ is a product as well as an intersection of finitely many of these different types of prime ideals. Such factorizations, when they are irredundant, are shown to be unique except for the order of the factors. We also characterize the Leavitt path algebras $L$ in which every ideal admits such factorizations and also in which every ideal is one of these special type of ideals.

math.RA↗

arXiv2018

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 $L_{K}(E)$ 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 $Σ$-$V$ rings. As an application of our results, we answer an open question raised in \cite{JST} whether an exchange $Σ$-$V$ ring has bounded index of nilpotence.

math.RA↗

arXiv2015

On intersections of two-sided ideals of Leavitt path algebras

Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. Powers of a single ideal $I$ are considered and it is shown that the intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest graded ideal of $L$ contained in $I$. This leads to an analogue of Krull's theorem for Leavitt path algebras.

math.RA↗

arXiv2008

Regularity conditions for arbitrary Leavitt path algebras

We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field $K$. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph $E$: (1) $L_K(E)$ is von Neumann regular. (2) $L_K(E)$ is $π$-regular. (3) $E$ is acyclic. (4) $L_K(E)$ is locally $K$-matricial. (5) $L_K(E)$ is strongly $π$-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.

math.RA↗