Mercedes Siles Molina

Socle theory for Leavitt path algebras of arbitrary graphs

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, and it is shown that the line points generate the socle of a Leavit t path algebra, extending so the results for row-finite graphs in the previous paper (12) ( but with different methods). A concrete description of the socle of a Leavitt path algebra i s obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, shorthening significantly the original ones.

Chain conditions for Leavitt path algebras

Abstract In this paper we give necessary and sufficient conditions on a row-finite graph E so that the corresponding (not necessarily unital) Leavitt path K-algebra LK (E) is either artinian or noetherian from both a local and a categorical perspective. These extend the known results in the unital case to a much wider context. Besides the graph theoretic conditions, we provide in both situations isomorphisms between these algebras and appropriate direct sums of matrix rings over K or K[x, x –1].

Local rings of exchange rings

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).

Quotient rings and Fountain-Gould left orders by the local approach

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

Associative and Lie algebras of 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 Q respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients.

EXCHANGE MORITA RINGS

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 analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal.

Computing the maximal algebra of quotients of a Lie algebra

Abstract 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.

ORDERS IN RINGS WITH INVOLUTION

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, then every involution * : R →R can be extended to a (unique) involution on Q in such a way that (R, *) is a *-order in (Q, *). And conversely, every *-order in an involution ring (Q, *) with Q semiprime and coinciding with its socle is a Fountain–Gould order inQ.

STRONGLY NON-DEGENERATE LIE ALGEBRAS

Let A be a semiprime 2 and 3-torsion free non-commutative associative alge- bra. 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 annihi- lator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra A with involution and the Lie algebra SDer(A) of involution pre- serving derivations of A.

Using Steinberg algebras to study decomposability of Leavitt path algebras

Abstract Given an arbitrary graph E we investigate the relationship between E and the groupoid G E {G_{E}} . We show that there is a lattice isomorphism between the lattice of pairs ( H , S ) {(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 ) {G_{E}^{(0)}} . We use this lattice isomorphism to characterise the decomposability of the Leavitt path algebra L K ⁢ ( E ) {L_{K}(E)} , where K is a field. First we find a graph condition to characterise when an open invariant subset of G E ( 0 ) {G_{E}^{(0)}} is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to L K ⁢ ( E ) {L_{K}(E)} being decomposable in the sense that it can be written as a direct sum of two nonzero ideals. We end by relating decomposability of a Leavitt path algebra with the existence of nontrivial central idempotents. In fact, all the nontrivial central idempotents can be described.