Related papers: Graded cancellation properties of graded rings and…

Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and…

Let $E$ be a directed graph, $\mathbb K$ be a field, and $\mathbb F$ be the free group on the edges of $E$. In this work, we use the isomorphism between Leavitt path algebras and partial skew group rings to endow $L_{\mathbb K}(E)$ with an…

Let $S$ be a partial groupoid, that is, a set with a partial binary operation. An $S$-graded ring $R$ is said to be graded von Neumann regular if $x\in xRx$ for every homogeneous element $x\in R.$ Under the assumption that $S$ is…

If $K$ is a field with involution and $E$ an arbitrary graph, the involution from $K$ naturally induces an involution of the Leavitt path algebra $L_K(E).$ We show that the involution on $L_K(E)$ is proper if the involution on $K$ is…

Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple,…

In this paper, we give a complete characterization of Leavitt path algebras which are graded $\Sigma $-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we…

Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no…

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a…

While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field,…

In this paper, we develop structure theory for graded regular graded self-injective rings and apply it in the context of Leavitt path algebras. We show that for a finite graph, graded regular graded self-injective Leavitt path algebras are…

In this paper we study the graded version of Naimark's problem for Leavitt path algebras considering them as $\mathbb{Z}$-graded algebras. Several characterizations are obtained of a Leavitt path algebra $L$ of an arbitrary graph $E$ over a…

This survey of the recent developments in the investigations of a Leavitt path algebra L of an arbitrary graph E over a field K consists of two parts. In the first part describes how very often a single graph property of E implies multiple…

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…

Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewed as a $G$-graded ring. In this article,…

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a…

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 provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields…

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…

We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we…

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a…