Leavitt path algebras with coefficients in a commutative ring

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 $L_K(E)\cong K{\otimes }_{\mathbb{Z}}{L}_{\mathbb{Z}}(E)$.

26 pages, Version II comments: numerous typos corrected, changes made to exposition, an error in Lemma 6.1 corrected, the statement of Theorem 8.1 modified; Version III comments: typos corrected, more references added, this is the version to be published