On the ideals of ultragraph Leavitt path algebras

TL;DR

This paper characterizes the structure of ideals in ultragraph Leavitt path algebras, providing explicit generators and classifying prime and primitive ideals, thereby advancing understanding of their algebraic properties.

Contribution

It offers a complete description of ideals, including generators, and characterizes prime and primitive ideals, proving Exel's Effros-Hahn conjecture in this context.

Findings

Explicit generators for all ideals provided

Complete classification of prime and primitive ideals

Proof of Exel's Effros-Hahn conjecture for ultragraph Leavitt path algebras

Abstract

In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen simple module. Consequently, we prove Exel's Effros-Hahn conjecture on primitive ideals in the ultragraph Leavitt path algebra setting (a conclusion that is also new in the context of Leavitt path algebras of graphs).