Ideal structure of Leavitt path algebras with coefficients in a unital commutative ring

TL;DR

This paper investigates the ideal structure of Leavitt path algebras over unital commutative rings, providing new characterizations of prime and primitive ideals, especially under Condition (K) when the coefficient ring is a field.

Contribution

It introduces modified definitions of basic ideals and characterizes prime and primitive ideals of Leavitt path algebras over arbitrary unital commutative rings.

Findings

Characterization of prime and primitive ideals in Leavitt path algebras.

Conditions for primeness and primitivity of $ l$.

Equivalence of prime and primitive ideals when $E$ satisfies Condition (K) and $R$ is a field.

Abstract

Let $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative ring. We analyze the ideal structure of the Leavitt path algebra $\lr$ introduced by Mark Tomforde. We first modify the definition of basic ideals and we then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of $\lr$. Then by applying these results we determine prime graded basic ideals and left (or right) primitive graded ideals of $\lr$. In particular, we show that when $E$ satisfies Condition (K) and $R$ is a field, the set of prime ideals and the set of primitive ideals of $\lr$ coincide.