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.     

The Commuting Graph of Minimal Nonsolvable Groups

The purpose of this paper is to prove that if G is a finite minimal nonsolvable group (i.e. G is not solvable but every proper quotient of G is solvable), then the commuting graph of G has diameter 3. We give an example showing that this result is the best possible. This result is related to the structure of finite quotients of the multiplicative group of a finite-dimensional division algebra.     

Invariant Subspaces for Commuting Operators on a Real Banach Space

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

$${\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,$$

where ║ · ║ _e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.

     

Real Commuting Differential Operators Connected with Two-Dimensional Abelian Varieties

We indicate smooth real commuting differential operators whose eigenvalues and eigenfunctions are parametrized by principally polarized abelian varieties.     

Real Commuting Differential Operators Connected with Two-Dimensional Abelian Varieties

We indicate smooth real commuting differential operators whose eigenvalues and eigenfunctions are parametrized by principally polarized abelian varieties.     

Some Properties of the Idempotent Graph of a Ring

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper, we show that diam\({(I(M_n(D))) = 4}\), for all natural number \({n \geq 4}\) and diam\({(I(M_3(D))) = 5}\), where D is a division ring. We also provide some classes of rings whose idempotent graphs are connected. Moreover, the regularity, clique number and chromatic number of idempotent graphs are studied.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.     

The Compressed Annihilator Graph of a Commutative Ring

Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R. The compressed annihilator graph of R is the graph AG_E(R), whose vertices are equivalence classes of zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(x)∪ann(y) ? ann(xy). For a reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.     

Note on Integral Distances

A planar point set S is called an integral set if all the distances between the elements of S are integers. We prove that any integral set contains many collinear points or the minimum distance should be relatively large if |S| is large.     

多圆盘上的交换对偶Toeplitz算子

We completely characterize commutativity of Sρ and Sψ on L2α (Dn)⊥ for bounded pluriharmonic symbols ρ and ψ on Dn, and prove that SρSψ=Sψρ if and only if ρ is analytic or ψ is analytic.     

Commuting Toeplitz operators on the Segal-Bargmann space

Consider two Toeplitz operators Tg, Tf on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily.     

Commuting Dual Toeplitz Operators on the Polydisk

On the polydisk, the commutativity of dual Toeplitz operators is studied. We obtain characterizations of commuting dual Toeplitz operators, essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.     

Commuting Toeplitz operators on the Dirichlet space

In this paper, we study the commutativity of Toeplitz operators with continuous symbols on the Dirichlet space. First, under a mild condition concerning absolute continuity we characterize (semi-)commuting Toeplitz operators. This is a generalization of the case of harmonic symbols. Also, if one of the symbol is radial or analytic, we get another characterization, which is different from the case on the Bergman space.     

The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal

For a commutative ring R with identity, the annihilating-ideal graph of R, denoted 𝔸𝔾(R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted 𝔸𝔾 _I (R). We discuss when 𝔸𝔾 _I (R) is bipartite. We also give some results on the subgraphs and the parameters of 𝔸𝔾 _I (R).     

Commuting maps on lie ideals

Let R be a ring and let A be a subset of R. A map f : A→ R is commuting on A if [f(x),x]= 0 for all xεA where [x,y] = xy — yx. Suppose that R is a prime ring of characteristic ≠2 with extended centroid C. If L is a noncommutative Lie ideal of R and f:L→R an additive commuting map, then there is λε C and an additive map ∈: L→ C such that f(v) = λ(v)=λv+∈((v) for all vεL.     

The Real Spectrum of a Noncommutative Ring and the Artin-Lang Homomorphism Theorem

Let R be a noncommutative ring. Two epimorphisms

$$\alpha_{i}:R\to (D_{i},\leqslant_{i}),\quad i = 1,2 $$

from R to totally ordered division rings are called equivalent if there exists an order-preserving isomorphism ? : (D ₁, ? ₁) → (D ₂, ? ₂) satisfying ? ° α ₁ = α ₂. In this paper we study the real epi-spectrum of R, defined to be the set of all equivalence classes (with respect to this relation) of epimorphisms from R to ordered division rings. We show that it is a spectral space when endowed with a natural topology and prove a variant of the Artin-Lang homomorphism theorem for finitely generated tensor algebras over real closed division rings.

     

Commuting Toeplitz Operators on the Pluriharmonic Bergman Space

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.     

About Distances between Points on the Plane

Mathematical Notes -