Chains of Semiprime and Prime Ideals in Leavitt Path Algebras

TL;DR

This paper characterizes semiprime ideals in Leavitt path algebras, explores their lattice structure, and confirms Kaplansky's conjecture for these algebras, advancing understanding of their ideal theory.

Contribution

It provides a generator-based description of semiprime ideals, reveals their lattice properties, and verifies Kaplansky's conjecture within Leavitt path algebras.

Findings

Semiprime ideals form a complete sublattice of all ideals.

Prime spectra of Leavitt path algebras have specific order and gap properties.

Kaplansky's conjecture holds for Leavitt path algebras.

Abstract

Semiprime ideals of an arbitrary Leavitt path algebra L are described in terms of their generators. This description is then used to show that the semiprime ideals form a complete sublattice of the lattice of ideals of L, and they enjoy a certain gap property identified by Kaplansky in prime spectra of commutative rings. It is also shown that the totally ordered sets that can be realized as prime spectra of Leavitt path algebras are precisely those that have greatest lower bounds on subchains and enjoy the aforementioned gap property. Finally, it is shown that a conjecture of Kaplansky regarding identifying von Neumann regular rings via their prime factors holds for Leavitt path algebras.