Leavitt path algebras with coefficients in a commutative ring

TL;DR

This paper extends the theory of Leavitt path algebras to coefficients in a commutative ring, providing new theorems and analyzing their ideal structures, generalizing classical results over fields.

Contribution

It introduces a construction for Leavitt path algebras over commutative rings and proves generalized versions of key theorems, simplifying the classical proofs.

Findings

Generalized Graded Uniqueness Theorem

Generalized Cuntz-Krieger Uniqueness Theorem

Isomorphism $L_K(E) \\cong K \\otimes_\\Z L_\\Z(E)$ for fields

Abstract

Given a directed graph E we describe a method for constructing a Leavitt path algebra $L_R(E)$ whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of $L_R(E)$, and we prove that if $K$ is a field, then $L_K(E) \cong K \otimes_\Z L_\Z(E)$.