Related papers: Every finitely generated ideal of a Leavitt path a…

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. We show that $L_K(E)$ is a B\'{e}zout ring, i.e., that every finitely generated one-sided ideal of $L_K(E)$ is…

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the two-sided ideals of L form a distributive lattice. It is also shown that L is a multiplication ring, that…

Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we…

Given an arbitrary graph E and a field K, the prime ideals as well as the primitive ideals of the Leavitt path algebra L_K(E) are completely described in terms of their generators. The stratification of the prime spectrum of L_K(E) is…

Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies…

Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. The necessary and sufficient con- ditions are given to assure the existence of a maximal ideal in $L$ and also the necessary…

We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a field $K$ as a direct sum of matrix rings over $K$, direct sum with a corner of the Leavitt…

In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for…

Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Pr\"ufer…

Using the E-algebraic systems, various graded irreducible representations of a Leavitt path algebra L of a graph E over a field K are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left…

We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the…

Let $E$ be an arbitrary graph and $K$ be any field. We construct various classes of non-isomorphic simple modules over the Leavitt path algebra $L_{K}(E)$ induced by vertices which are infinite emiters, closed paths which are exclusive…

Let E be an arbitrary directed graph with no restrictions on the number of vertices and edges and let K be any field. We give necessary and sufficient conditions for the Leavitt path algebra L_K(E) to be of countable irreducible…

In this paper, we classify all Leavitt path algebras which have the property that every Lie ideal is an ideal. As an application, we show that Leavitt path algebras with this property provide a class of locally finite, infinite-dimensional…

For any row-finite graph $E$ and any field $K$ we construct the {\its Leavitt path algebra} $L(E)$ having coefficients in $K$. When $K$ is the field of complex numbers, then $L(E)$ is the algebraic analog of the Cuntz Krieger algebra…

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras L_K(E) that…

In this paper a bijection between the set of prime ideals of a Leavitt path algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient conditions…

There is an extensive recent literature on the graded, non-graded, prime, primitive, maximal ideals of Leavitt path algebras. In this introductory level survey, we will be giving an overview of different types of ideals and the…

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 underlying graph. When the algebra decomposes, it…

If $E$ is a graph and $K$ is a field, we consider an ideal $I$ of the Leavitt path algebra $L_K(E)$ of $E$ over $K$. We describe the admissible pair corresponding to the smallest graded ideal which contains $I$ where the grading in question…