Mercedes Siles Molina

University of Malaga | UMA · Department of Algebra, Geometry and Topology

A ring $R$ is called right (or left) {\it socle-injective} if every $R$-homomor\-phism from the right (or left) socle of $R$ into $R$ extends to $R$. In this paper, we show that any semiprime ring $R$ with socle $S$, is socle-injective if and only if $\End_R(S) \cong Q'/I$, where $Q'$ is a suitable subring of maximal right ring of quotients of $R$...

We describe the centroid of some Leavitt path algebras. More precisely, we show that for Leavitt path algebras over a field $K$ that are simple its centroid is isomorphic to $K$, and for prime Leavitt path algebras its centroid is isomorphic to $K$ except if the graph is a row-finite comet, in which case the centroid is isomorphic to $K[x,x^{-1}]$.

We identify the largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purel...

We associate a square to any two-dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behavior of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic g...

In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator plus extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in non-degenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis...

In this paper we complete the classification of four dimensional perfect non-simple evolution algebras (under mild conditions on the based field), started in Casado et al. (Linear Algebra and its Applications, [1]). We consider the different parametric families of evolution algebras appearing in the classification and study which algebras in the sa...

This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is...

With the aim of finding useful tools and invariants to classify finite dimensional evolution algebras, we introduce and study the notion of a basic ideal. Every n-dimensional perfect evolution algebra has a maximal basic ideal I which is unique except when the dimension of I is n-1. An application of our results leads to the description of the four...

We identify largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely in...

We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic...

We classify the four dimensional perfect non-simple evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

The richness of the idempotent structure of Leavitt path algebras lies at the heart of the subject; in this chapter we present a number of topics which fall under this umbrella. These include: the purely infinite property (for both simple and non-simple algebras); the structure of the monoid of finitely generated projective modules; the exchange pr...

In this chapter we investigate many of the K-theoretic properties of LK(E). We start by considering the Grothendieck group K0(LK(E)), and then subsequently the Whitehead group K1(LK(E)). Next, we discuss one of the central currently-unresolved questions in the subject (the so-called Algebraic Kirchberg Phillips Question) which asks whether certain...

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra LK(E) for an arbitrary directed graph E and field K. After providing some basic examples, we show how Leavitt path algebras are related to the monoid realization alg...

In this chapter we provide descriptions of Leavitt path algebras satisfying various well-studied ring-theoretic properties. These include: primeness and primitivity; chain conditions on one-sided ideals; self-injectivity; and the stable rank.

We conclude the book with various observations regarding three important aspects of Leavitt path algebras. First, we describe various generalizations of, and constructions related to, Leavitt path algebras. Next, we present some applications of Leavitt path algebras (specifically, we give some examples of results from outside the subject of Leavitt...

In this chapter we investigate the ideal structure of Leavitt path algebras. We start by describing the natural \(\mathbb{Z}\)-grading on LK(E). We then present the Reduction Theorem; this result describes how elements of LK(E) may be transformed in some specified way to either a vertex or a cycle without exits. Numerous consequences are discussed,...

In this chapter we investigate the connections between Leavitt path algebras (with coefficients in \(\mathbb{C}\)), and their analytic counterparts, the graph C∗-algebras. We start by giving a brief overview of graph C∗-algebras, and then show how the Leavitt path algebra \(L_{\mathbb{C}}(E)\) naturally embeds as a dense ∗-subalgebra of the graph C...

We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the $K_0$ group, $\det(N'_E)$ (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated...

We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the $K_0$ group, $\det(N'_E)$ (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated...

We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path...

We classify three dimensional evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

We classify three dimensional evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group t...

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subs...

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subs...

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subs...

Given an arbitrary graph $E$ we investigate the relationship between $E$ and the groupoid $G_E$. We show that there is a lattice isomorphism between the lattice of pairs $(H, S)$, where $H$ is a hereditary and saturated set of vertices and $S$ is a set of breaking vertices {associated to $H $}, onto the lattice of open invariant subsets of $G_E^{(0...

Given an arbitrary graph $E$ we investigate the relationship between $E$ and the groupoid $G_E$. We show that there is a lattice isomorphism between the lattice of pairs $(H, S)$, where $H$ is a hereditary and saturated set of vertices and $S$ is a set of breaking vertices {associated to $H $}, onto the lattice of open invariant subsets of $G_E^{(0...

We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also prove the existence and unicity of a direct sum decomposition into irreducible components for every non-degene...

In this paper we prove that if two idempotent rings R and S are Morita equivalent then for every von Neumann regular element a ∈ R the local algebra of R at a, R a , is isomorphic to \(\mathbb{M}_{n}(S)_{u}\) for some natural n and some idempotent u in \(\mathbb{M}_{n}(S)\). We give examples showing that the converse of this result is not true in g...

In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the ke...

Presenting the collaborations of over thirty international experts in the latest developments in pure and applied mathematics, this volume serves as an anthology of research with a common basis in algebra, functional analysis and their applications. Special attention is devoted to non-commutative algebras, non-associative algebras, operator theory...

We investigate conditions under which the endomorphism ring of the Leavitt path algebra $L_{K}(E)$ possesses various ring and module-theoretical properties such as being von Neumann regular, $\pi$-regular, strongly $\pi$-regular or self-injective. We also describe conditions under which $L_{K}(E)$ is continuous as well as automorphism invariant as...

Let E be an arbitrary graph, and let K be any field. We show that many generalized regularity conditions for the Leavitt path algebra L K (E) are equivalent and that this happens exactly when the graph E satisfies Condition (K).

In this paper we prove that two idempotent rings are Morita equivalent if every corner of one of them is isomorphic to a corner of a matrix ring of the other one. We establish the converse (which is not true in general) for $\sigma$-unital rings having a $\sigma$-unit consisting of von Neumann regular elements. The following aim is to show that a p...

In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the ke...

We characterize, in terms of its idempotents, the Leavitt path algebras of an arbitrary graph that satisfies Condition (L) or Condition (NE). In the latter case, we also provide the structure of such algebras. Dual graph techniques are considered and demon-strated to be useful in the approach of the study of Leavitt path algebras of arbitrary graph...

We show that compact graph C ∗-algebras C ∗(E) are topological direct sums of finite matrices over ℂ and KL(H), for some countably dimensional Hilbert space, and give a graph-theoretic characterization as those whose graphs are row-finite, acyclic and every infinite path ends in a sink. We further specialize in the simple case providing both struct...

We show that the algebra Der(L)Der(L) of derivations of a strongly nondegenerate Lie algebra L graded by an ordered group G with a finite grading (and satisfying a mild technical condition) inherits the grading from L, i.e., Der(L)Der(L), which turns out to be a strongly nondegenerate Lie algebra, is G-graded and its support has the same length as...

The motivation for this paper has been to study the relation between the zero component of the maximal graded algebra of quotients and the maximal graded algebra of quotients of the zero component, both in the Lie case and when considering Martindale algebras of quotients in the associative setting. We apply our results to prove that the finitary c...

We study the center of several types of path algebras. We start with the path algebra $KE$ and prove that if the number of vertices is infinite then the center is zero. Otherwise, it coincides with the field $K$ except when the graph $E$ is a cycle in which case the center is $K[x]$, the polynomial algebra in one indeterminate. Then we compute the...

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We...

The aim of this work is the description of the isomorphism classes of all Leavitt path algebras coming from graphs satisfying Condition (Sing) with up to three vertices. In particular, this classification recovers the one achieved by Abrams et al. in the case of graphs whose Leavitt path algebras are purely infinite simple. The description of the i...

In this paper we develop a Fountain–Gould-like Goldie theory for alternative rings. We characterize alternative rings which are Fountain–Gould left orders in semiprime alternative rings coinciding with their socle, and those which are Fountain–Gould left orders in semiprime artinian alternative rings.

We characterize the Leavitt path algebras over arbitrary graphs which are weakly regular rings as well as those which are self-injective. In order to reach our goals we extend and prove several results on projective, injective and flat modules over Leavitt path algebras and, more generally, over (not necessarily unital) rings with local units. Key...

In this paper we explore graded algebras of quotients of Lie algebras with special emphasis on the 3-graded case and answer some natural questions concerning its relation to maximal Jordan systems of quotients.

In this paper, results known about the artinian and noetherian conditions for the Leavitt path algebras of graphs with finitely many vertices are extended to all row-finite graphs. In our first main result, necessary and sufficient conditions on a row-finite graph E are given so that the corresponding (not necessarily unital) Leavitt path K-algebra...

In this paper we explore graded algebras of quotients of Lie algebras with special emphasis on the 3-graded case and answer some natural questions concerning its relation to maximal Jordan systems of quotients.

The maximal algebra of quotients of a semiprime Lie algebra was introduced recently by M. Siles Molina. In the present paper we answer some natural questions concerning this concept, and describe maximal algebras of quotients of certain Lie algebras that arise from associative algebras.

We investigate the ascending Loewy socle series of Leavitt path algebras $L_K(E)$ for an arbitrary graph $E$ and field $K$. We classify those graphs $E$ for which $L_K(E)=S_{\lambda}$ for some element $S_{\lambda}$ of the Loewy socle series. We then show that for any ordinal $\lambda$ there exists a graph $E$ so that the Loewy length of $L_K(E)$ is...

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such el...

In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equiva...

Let A be a semiprime 2 and 3-torsion free non-commutative associative algebra. We show that the Lie algebra Der(A) of(associative) derivations of A is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of A. This result follows from a description of the quadratic annih...

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points,...

In this paper a bijection between the set of prime ideals of a Leavitt path algebra LK(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 on the graph E so that the Leavitt path algebra LK(E) is primitive are also found. introduction Leavitt path algebras o...

A group-graded K-algebra A = ⊕g∈G A g is called locally finite in case each graded component A g is finite dimensional over K. We characterize the graphs E for which the Leavitt path algebra L K (E) is locally finite in the standard ℤ-grading. For a locally finite ℤ-graded algebra A we show that, if every nonzero graded ideal has finite codimension...

We classify the directed graphs E for which the Leavitt path algebra L(E) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, modulo the one-dimensional ideals, all finite dimensional Leavitt path algebras arise.

Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra has essential socle). We also int...

In this paper we characterize the minimal left ideals of a Leavitt path algebra as those which are isomorphic to principal left ideals generated by line points; that is, by vertices whose trees contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line poi...

In this paper we characterize the minimal left ideals of a Leavitt path algebra as those ones which are isomorphic to principal left ideals generated by line point vertices, that is, by vertices whose trees do not contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generat...

Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra has essential socle). We also int...

We show that every finite Z-grading of a simple associative algebra A comes from a Peirce decomposition induced by a complete system of orthogonal idempotents lying in the maximal left quotient algebra of A (which coincides with the graded maximal left quotient algebra of A). Moreover, a nontrivial 3-grading can be found. This grading provides 3-gr...

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms (of left A–modules) from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the co...

In this paper we introduce a definition of order in a (notnecessarily unital) ring with involution in terms of the notions of Moore–Penrose inverse and *-cancellable element instead of those of group inverse and cancellable element. The main result states that if R is a Fountain–Gould order in a ring Q with Q semiprime and coinciding with its socle...

In this paper, we introduce a notion of weak Fountain–Gould left order for associative pairs and give a Goldie–like theory of associative pairs which are weak Fountain–Gould left orders in semiprime pairs coinciding with their socles.

In this paper we construct the maximal left quotient system of every pair of modules in some generalized matrix rings. These rings can be seen as 3-graded algebras or as superalgebras. We show the relation among the three different notions of left quotients.

In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of quotients of a Lie algebra $L$ in terms of the associative algebras generated by the adjoint operators of $L$ and...

We characterize Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 or $\infty$. Concrete criteria in terms of properties of the underlying graph are given for each case.

We prove that under conditions of regularity the maximal left quotient ring of a corner of a ring is the corner of the maximal left quotient ring. We show that if R and S are two non-unital Morita equivalent rings then their maximal left quotient rings are not necessarily Morita equivalent. This situation contrasts with the unital case. However we...

In this paper, we prove that left nonsingularity and left nonsingularity plus finite left local Goldie dimension are two Morita invariant properties for idempotent rings without total left or right zero divisors. Moreover, two Morita equivalent idempotent rings, semiprime and left local Goldie, have Fountain-Gould left quotient rings that are Morit...

We investigate the basic properties of the different socles that can be considered in not necessarily semiprime associative systems. Among other things, we show that the socle defined as the sum of minimal (or minimal and trivial) inner ideals is always an ideal. When trivial inner ideals are included, this inner socle contains the socles defined i...

In this paper we introduce the notion of algebra of quotients of a Lie algebra. Properties such as semiprimeness, primeness or nondegeneracy can be lifted from a Lie algebra to its algebras of quotients. We construct a maximal algebra of quotients for every semiprime Lie algebra and give a Passman-like characterization of this (unique) maximal alge...

We study Fountain-Gould left orders in semiprime rings coinciding with their socles by means of local rings at elements.

In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an...

We develop a Goldie theory for associative pairs and characterize associative pairs which are orders in semiprime associative pairs coinciding with their socle, and those which are orders in semiprime artinian associative pairs

We characterize the exchange property for non-unital rings in terms of their local rings at elements,and we use this characterization to show that the exchange property is Morita invariant for idempotent rings.We also prove that every ring contains a greatest exchange idela(with respect to the inclusion).

The usual Jordan canonical form for matrices is extended first to nilpotent elements of the socle of a nondegenerate Jordan algebra and then to elements of a nondegenerate Jordan algebra which is reduced over an algebraically closed field.

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element aJ has a (unique) generalized inverse if and only if it is strongly regular, i.e., aP(a)2J. A Jordan...

The theory of associative algebras of quotients has a rich history and is still an active research area. In the recent paper [24], the author iniciated the study of algebras of quotients in the Lie setting and built a maximal algebra of quotients for every semiprime Lie algebra. The inspiration comes from the associative ([25]) and Jordan ([18]) co...