On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.     

Commuting Graphs of Group Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.     

Planar,Toroidal, and Projective Commuting and Noncommuting Graphs

In this paper, all finite groups whose commuting (noncommuting) graphs can be embed on the plane, torus, or projective plane are classified.     

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

Matrix Completions and Chordal Graphs

In a matrix-completion problem the aim is to specifiy the missing entries of a matrix in order to produce a matrix with particular properties. In this paper we survey results concerning matrix-completion problems where we look for completions of various types for partial matrices supported on a given pattern. We see that the existence of completions of the required type often depends on the chordal properties of graphs associated with the pattern.     

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Let R ? ? be a GCD-domain. In this article, Weinberg's conjecture on the n × n matrix algebra M _n (R) (n ≥ 2) is proved. Moreover, all the lattice orders (up to isomorphisms) on a full 2 × 2 matrix algebra over R are obtained.     

Commuting graphs of groups and related numerical parameters

Given a finite group G, we denote by Δ(G) the commuting graph of G which is defined as follows: the vertex set is G and two distinct vertices x and y are joined by an edge if and only if xy = yx. Clearly, Δ(G) is always connected for any group G. We denote by κ(G) the number of spanning trees of Δ(G). In the present paper, among other results, we first obtain the value κ(G) for some specific groups G, such as Frobenius groups, Dihedral groups, AC-groups, etc. Next, we characterize the alternating group A₅, in the class of nonsolvable groups through its tree-number κ(A₅). Finally, we classify the finite groups for which the power graph and the commuting graph coincide.     

On diameter of the commuting graph of a full matrix algebra over a finite field

It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja [Linear Algebra Appl. 418 (2006) 161–176].     

m-Power Commuting MAPS on Semiprime Rings

Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 ? e)U ? M₂(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting.     

Matrix graphs and MRD codes over finite principal ideal rings

Let R be a finite principal ideal ring and $m,n,d$ positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are $m\times n$ matrices over R and two matrices A and B are adjacent if and only if $0<\mathrm{rank}(A-B)<d$. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.     

On the Commuting Graph of Dihedral Group

Let Γ be a non-abelian group and Ω ? Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D_2n. We also study the metric dimension of commuting graph on D_2n and compute its resolving polynomial.     

Sets of Commuting Derivations and Simplicity

A ring R is simple under a set D of derivations if no nontrivial ideal of R is preserved by all derivations in D. Continuing previous joint work with C. J. Maxson, the author provides a computational test for the simplicity of k[x ₁,…,x _n ]/〈 x ₁ ^p ,…, x _n ^p 〉 (k a field of characteristic p > 0) under a set of commuting k-derivations. Specific rings are then examined for sets of commuting derivations, especially those under which the ring is simple. The possible sizes and minimality of such sets are also determined in particular cases.     

Commuting pairs and triples of matrices and related varieties

In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M_n(F), and show that it is irreducible of dimension n²−n for n4, but reducible, of dimension greater than n²−n for n7.     

On Commuting Graph of Group Ring $Z_nS_3$

The commuting graph of an arbitrary ring $R$, denoted by $Γ(R)$, is a graph whose vertices are all non-central elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $ab = ba$. In this paper, we investigate the connectivity and the diameter of $Γ(Z_n S_3)$. We show that $Γ(Z_n S_3)$ is connected if and only if $n$ is not a prime number. If $Γ(Z_n S_3)$ is connected then diam $(Γ(Z_n S_3)) = 3$, while if $Γ(Z_n S_3)$ is disconnected then every connected component of $Γ(Z_n S_3)$ must be a complete graph with same size, and we completely determine the vertice set of every connected component.     

Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

The article gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from x to y is equal to the complex unity i (and its symmetric entry is ) if the reverse arc is not present. We also allow arcs in both directions and unoriented edges, in which case we use 1 as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced subdigraphs. Besides covering the basic properties, we discuss many differences from the properties of eigenvalues of undirected graphs and develop basic theory. The main novel results include the following. Several surprising facts are discovered about the spectral radius; some consequences of the interlacing property are obtained; operations that preserve the spectrum are discussed—they give rise to a large number of cospectral digraphs; for every , all digraphs whose spectrum is contained in the interval are determined.     

矩阵尾环的Morphic性质(英文)

本文的主要目的是考虑强Morphic环D上的矩阵尾环R[D]的Morphic性质。本文讨论了类似尾环的一些性质。证明了:R[D]是强左Morphic环当且仅当R[D]是左Morphic环当且仅当D是强左Morphic环。本文还构造了一些例子来说明问题。     

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999 Nicholson , W. K. (1999). Strongly clean rings and Fitting's lemma. Comm. Algebra 27:3583–3592. [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.     

The Total Graphs of Finite Rings

In this paper we extend the study of total graphs τ(R) to noncommutative finite rings R. We prove that τ(R) is connected if and only if R is not local, and we see that in that case τ(R) is always Hamiltonian. We also find an upper bound for the domination number of τ(R) for all finite rings R.     

Lattice-Ordered Matrix Rings Over the Integers

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.