Unital Ring Research Articles

Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

We show that any pointed, preordered module map BFgr(E)→BFgr(F) between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving ⁎-homomorphism Lℓ(E)→Lℓ(F) between the corresponding Leavitt path algebras over any commutative unital ring with involution ℓ. Specializing to the case when ℓ is a field, we establish the fullness part of Hazrat's conjecture about the functor from Leavitt path ℓ-algebras of finite graphs to preordered modules with order unit that maps Lℓ(E) to its graded Grothendieck group. Our construction of lifts is of combinatorial nature; we characterize the maps arising from this construction as the scalar extensions along ℓ of unital, graded ⁎-homomorphisms LZ(E)→LZ(F) that preserve a sub-⁎-semiring introduced here.

Reticulation of an integral complete l-groupoid: An axiomatic approach

The reticulation of a unital ring R is a bounded distributive lattice L(A) whose main property is that the Zariski prime spectrum of R is homeomorphic with the Stone prime spectrum of L(A). In this paper we develop an axiomatic theory of reticulation in the abstract framework offered by the integral complete l-groupoids (= icl-groupoids). For an algebraic icl-groupoid A we define axiomatically three basic notions: pre-reticulation, semi-reticulation and reticulation. We prove a uniqueness theorem for the semi-reticulations of A and we characterize the algebraic icl-groupoids that admit a reticulation. Given an algebraic icl-groupoid A, we define the spectral closure of the m-prime spectrum SpecZ(A) of A, an abstraction of the spectral closure of the prime spectrum of a ring. When A is the icl-groupoid Id(R) of ideals in a ring R we obtain some of the Belluce results on the reticulation of R and the spectral closure of the prime spectrum of R.

A necessary and sufficient condition for the existence of nontrivial Sn-invariants in the splitting algebra

For a monic polynomial [Formula: see text] over a commutative, unitary ring [Formula: see text] the splitting algebra [Formula: see text] is the universal [Formula: see text]-algebra such that [Formula: see text] splits in [Formula: see text]. The symmetric group acts on the splitting algebra by permuting the roots of [Formula: see text]. It is known that if the intersection of the annihilators of the elements [Formula: see text] and [Formula: see text] (where [Formula: see text] depends on [Formula: see text]) in [Formula: see text] is zero, then the invariants under the group action are exactly equal to [Formula: see text]. We show that the converse holds.

Leavitt path algebras of labelled graphs

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.

Systems of divided powers in algebras of multivariate Hurwitz series

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus.

A note on generalized Jordan n-derivations of unital rings

A note on generalized Jordan n-derivations of unital rings

High dimensional expanders and coset geometries

High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant.In this work, we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. This property has a number of important consequences, including geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. Our construction also yields new families of expander graphs which are close to the Ramanujan bound, i.e., their spectral gap is close to optimal.The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved previously known construction of the Ramanujan complexes. The construction is also very symmetric (such symmetry properties are not known for Ramanujan complexes) ; The symmetry of the construction could be used, for example, in order to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.The main tool that we use for is the theory of coset geometries. Coset geometries arose as a tool for studying finite simple groups. Here, we show that coset geometries arise in a very natural manner for groups of elementary matrices over any finitely generated algebra over a commutative unital ring. In other words, we show that such groups act simply transitively on the top dimensional face of a pure, partite, clique complex.

SIMPLICITY OF LEAVITT PATH ALGEBRAS VIA GRADED RING THEORY

AbstractSuppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using 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, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

Universal cohomology theories

We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in R-linear abelian categories of motivic nature, where R is any commutative unitary ring. Universal homology theory on the one point category yields “hieratic” R-modules, i.e. the indization of Freyd’s free abelian category on R. Grothendieck -functors and satellite functors are recovered as certain additive relative homologies on an abelian category for which we also show the existence of universal ones.

Malcolmson semigroups

Inspired by the construction of the Cuntz semigroup for a C⁎-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C⁎-algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimension functions.

Classification of 1-absorbing comultiplication modules over a pullback ring

One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

Krull-Remak-Schmidt decompositions in Hom-finite additive categories

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A Hom-finite category is an additive category A for which there is a commutative unital ring k, such that each Hom-set in A is a finite length k-module. The aim of this note is to provide a proof that a Hom-finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.

Additivity of nonlinear higher anti-derivable mappings on generalized matrix algebras

<abstract><p>In this article, we proved that each nonlinear higher anti-derivable mapping on generalized matrix algebras is automatically additive. As for its applications, we find a similar conclusion on triangular algebras, full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras and factor von Neumann algebras respectively.</p></abstract>

Presentations of Munn matrix algebras over K-algebras with K being a commutative ring

We consider the Munn matrix algebras over anassociative unitalK-algebraA, whereKis a commutative (unital)ring andAas aK-module is free (of őnite or inőnite rank), and,for each (not necessarily őnitely deőned) presentation ofA, we givepresentations of the Munn matrix algebras over it.

Multiplicative Jordan type higher derivations of unital rings with non trivial idempotents

Multiplicative Jordan type higher derivations of unital rings with non trivial idempotents

Commutative unital rings elementarily equivalent to prescribed product rings

Commutative unital rings elementarily equivalent to prescribed product rings

Lie algebra of column-finite infinite matrices: Ideals and derivations

In this paper, we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers N. We describe the lattice of its ideals for an arbitrary field K and study its derivations over any commutative, unital ring R.

On generators of incidence rings over finite posets

For a matrix incidence algebra over a field, there is a full description of its generating subsets. We generalise this result to the case of matrix incidence rings over an arbitrary unital associative ring. Generating subsets of the following type are under consideration: if one adds all scalar matrices to the subset, then this greater set will generate the incidence ring. We obtain the full characterisation of such subsets. Moreover, their minimum cardinality is calculated precisely when the ring of coefficients is simple or semilocal. In order to obtain these results we introduce the graph of full differences of a ring and study its clique number.

On the essential algebra of the shifted Burnside biset functor

We describe the essential algebra, kBTˆ(G), of the Burnside biset functor shifted by a group T, at a group G, in two cases. First, when G and T are both finite abelian groups and k is a field of characteristic 0. In this case, kBTˆ(G) is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of G×G×T. The second case is when G and T are any finite groups satisfying (|G|,|T|)=1 and k is a commutative unitary ring. In this case, kBTˆ(G) is isomorphic to a semidirect product of Out(G) and kBZ(G)(T), the monomial Burnside ring of T with coefficients in Z(G).The aim of the article is to consider the natural set of generators of kBTˆ(G) coming from the transitive elements in kBT(G×G) and explore some cases in which it is possible to give a basis for kBTˆ(G) in this set.

Multiplicative anti-derivations of generalized n-matrix rings

In this paper, we prove that every multiplicative anti-derivation map of an arbitrary generalized [Formula: see text]-matrix ring is additive. Even more, as consequence, we get the same conclusion for the class of triangular [Formula: see text]-matrix rings, unital prime rings with a nontrivial idempotent, standard operator algebras and factor von Neumann algebras.