Variations of primeness and Factorization of ideals in Leavitt Path Algebras

TL;DR

This paper explores various types of prime ideals in Leavitt path algebras, providing conditions for their factorizations and characterizing algebras where all ideals have such properties.

Contribution

It introduces three new variations of prime ideals in Leavitt path algebras and establishes criteria for their factorizations and uniqueness.

Findings

Characterization of when ideals are products or intersections of prime ideals.

Conditions for unique factorizations of ideals.

Identification of Leavitt path algebras where all ideals are of these prime types.

Abstract

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.