Related papers: On Trace Zero Matrices and Commutators

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall…

Let $\mathscr{R}$ be a prime ring of Char$(\mathscr{R}) \neq 2$ and $m\neq 1$ be a positive integer. If $S$ is a nonzero skew derivation with an associated automorphism $\mathscr{T}$ of $\mathscr{R}$ such that $([S([a, b]), [a, b]])^{m} =…

In the present paper it is proved that every inner 2-local derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over a commutative associative ring $\Re$ is an inner derivation. Also, it is proved that, every derivation on an…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…

In this paper we study algebraic sets of pairs of matrices defined by the vanishing of either the diagonal of their commutator matrix or its anti-diagonal. We find a system of parameters for the coordinate rings of these two sets and their…

Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for…

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$, ${\mathfrak q}\colon \mathcal{G}\times\mathcal{G}\longrightarrow \mathcal{G}$ be an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak…

We study the trace set of the commutator subgroup of $\Gamma(2),$ a type of Local-Global problem about thin groups. We determine the local obstructions and then use the correspondence between binary quadratic forms and hyperbolic matrices…

We show that for all $k\ge 1$, there exists an integer $N(k)$ such that for all $n\ge N(k)$ the $k$-th order jet scheme over the commuting $n\times n$ matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for…

In this paper, we present a new method for the analysis of piecewise dynamical systems that are similar to the Collatz conjecture in regard to certain properties of the commutator of their sub-functions. We use the fact that the commutator…

We propose a new class of traces motivated by a trace/trace class property discovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products of operators outside the trace class. Spectral traces, traces that depend only on the…

If a noncommutative polynomial $f$ is neither an identity nor a central polynomial of $\mathcal A=M_n(\C)$, then every trace zero matrix in $\mathcal A$ can be written as a sum of two matrices from $f(\mathcal A)-f(\mathcal A)$. Moreover,…

Given a ring $R$ with center $Z(R)$, we say a linear map $f:R\rightarrow R$ is commuting if $[f(x),x]=0$ for all $x\in R$. Such a map has a standard form if there exists $\lambda\in R$ and additive $\mu:R\rightarrow Z(R)$ such that…

Let R be a commutative noetherian local ring that is not Gorenstein. It is known that the category of totally reflexive modules over R is representation infinite, provided that it contains a non-free module. The main goal of this paper is…

Let $R$ be a commutative ring and $\Gamma(R)$ denote its zero-divisor graph. In this paper, we investigate the genus number of the compact Riemann surface which $\Gamma(R)$ can be embedded and illustrate all finite commutative rings $R$ (up…

Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that…

Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we…

Given two elements $a$ and $b$ of a noncommutative ring, we express $\left( ba\right)^n$ as a "row vector times matrix times column vector" product, where the matrix is the $n$-th power of a matrix with entries…

We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$…

A ring R is said to be of stable range 1.5 if for each a, b from R and nonzero c from R satisfying aR + bR + cR = R there exists r from R such that (a + br)R + cR = R. Let R be a commutative domain in which all finitely generated ideals are…