The Theory of Prime Ideals of Leavitt Path Algebras over Arbitrary Graphs

TL;DR

This paper provides a comprehensive algebraic and graphical characterization of prime and primitive ideals in Leavitt path algebras over arbitrary graphs, including their stratification, Krull dimension, and minimal primes.

Contribution

It offers a complete description of prime and primitive ideals, stratification, and Krull dimension conditions for Leavitt path algebras based on graph properties.

Findings

Prime and primitive ideals are fully described via generators.

Conditions for all prime ideals to be primitive are established.

Leavitt path algebras of Krull dimension zero are characterized.

Abstract

Given an arbitrary graph E and a field K, the prime ideals as well as the primitive ideals of the Leavitt path algebra L_K(E) are completely described in terms of their generators. The stratification of the prime spectrum of L_K(E) is indicated with information on its individual stratum. Necessary and sufficient conditions are given on the graph E under which every prime ideal of L_K(E) is primitive. Leavitt path algebras of Krull dimension zero are characterized and those with various prescribed Krull dimension are constructed. The minimal prime ideals of L_K(E) are are described in terms of the graphical properties of E and using this, complete descriptions of the height one as well as the co-height one prime ideals of L_K(E) are given.