# Leavitt path algebras with coefficients in a commutative ring

@article{Tomforde2009LeavittPA,
title={Leavitt path algebras with coefficients in a commutative ring},
author={Mark Tomforde},
journal={arXiv: Operator Algebras},
year={2009}
}
• M. Tomforde
• Published 2009
• Mathematics
• arXiv: Operator Algebras
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 $L_K(E) \cong K \otimes_\Z… Expand 76 Citations ON LEAVITT PATH ALGEBRAS OVER COMMUTATIVE RINGS • Mathematics • International Electronic Journal of Algebra • 2019 In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt pathExpand On the homological dimensions of Leavitt path algebras with coefficients in commutative rings • Mathematics • 2016 In this paper, we give sharp bounds for the homological dimensions of the Leavitt path algebra$L_K(E)$of a finite graph$E$with coefficients in a commutative ring$K$, as well as establish aExpand Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L R (E) introduced by Mark Tomforde. We first modify the definition of basicExpand Simpleness of Leavitt path algebras with coefficients in a commutative semiring • Mathematics • 2015 In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, establishing some fundamental properties ofExpand The commutative core of a Leavitt path algebra • Mathematics • 2015 For any unital commutative ring$R$and for any graph$E$, we identify the commutative core of the Leavitt path algebra of$E$with coefficients in$R$, which is a maximal commutative subalgebra ofExpand Leavitt path algebras with coefficients in a Clifford semifield • Mathematics • Communications in Algebra • 2019 Abstract In this article, we define the Leavitt path algebra of a directed graph Γ with coefficients in a Clifford semifield S. The general properties of are briefly discussed. Then, concentrating onExpand Decomposable Leavitt path algebras for arbitrary graphs • Mathematics • 2015 For any field$K$and for a completely arbitrary graph$E$, we characterize the Leavitt path algebras$L_K(E)$that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlyingExpand Leavitt path algebras with bases consisting solely of units • Mathematics • Journal of Algebra • 2019 Abstract An algebra is said to be an invertible algebra if it has a basis consisting solely of units. Given a field K and a finite graph E, we give a condition on E that is equivalent to that of theExpand Morita Equivalence of Graph and Ultragraph Leavitt Path Algebras The primary purpose of this thesis is to show every ultragraph Leavitt path algebra is Morita equivalent, as a ring, to a graph Leavitt path algebra. Takeshi Katsura, Paul Muhly, Aidan Sims, and MarkExpand Derivations of Leavitt path algebras Abstract In this paper, we describe the K-module H H 1 ( L K ( Γ ) ) of outer derivations of the Leavitt path algebra L K ( Γ ) of a row-finite graph Γ with coefficients in an associative commutativeExpand #### References SHOWING 1-10 OF 30 REFERENCES The Leavitt path algebra of a graph • Mathematics • 2005 Abstract For any row-finite graph E and any field K we construct the Leavitt path algebra L ( E ) having coefficients in K . When K is the field of complex numbers, then L ( E ) is the algebraicExpand Uniqueness theorems and ideal structure for Leavitt path algebras Abstract We prove Leavitt path algebra versions of the two uniqueness theorems of graph C ∗ -algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras andExpand Nonstable K-theory for Graph Algebras • Mathematics • 2004 We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property andExpand The Leavitt path algebras of arbitrary graphs • Mathematics • 2008 We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ring-theoretic properties of these more general structures relate to the correspondingExpand The$C^*\$-Algebras of Arbitrary Graphs
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