Prime and Primitive Ideals of Ultragraph Leavitt Path Algebras

TL;DR

This paper characterizes prime and primitive ideals in ultragraph Leavitt path algebras, providing a detailed description of their structure and identifying conditions for graded and non-graded primes.

Contribution

It offers a comprehensive description of prime and primitive ideals in ultragraph Leavitt path algebras, including graded and non-graded cases, expanding understanding of their ideal structure.

Findings

Graded prime ideals are characterized via downward directed sets.

Non-graded prime ideals are explicitly described.

Non-graded prime ideals are always primitive.

Abstract

Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we characterize the non-graded prime ideals. We show that the non-graded prime ideals of $L_K(\mathcal G)$ are always primitive.