Leavitt path algebras with coefficients in a commutative ring

Given a directed graph E we describe a method for constructing a Leavitt path algebra LR(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 LR(E), and we prove that if K is a field, then LK(E)≅K⊗ZLZ(E).