Related papers: $(k+1)$-potent Matrices in triangular matrix Group…

In the present article we shown a formula to compute the number of all matrices over the finite field $F$ whit prescribed eigenvalues. Using this formula we obtain one inequality for the number of $(k+1)$-potent elements over finite rings.

We study sums of $k$-potent matrices. We show the conditions by which a complex matrix $A$ can be expressed as a sums of $k$-potent matrices. Also we obtain conditions by which a complex matrix $A$ can be expressed as a sum of finite order…

We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic…

Let $R$ be an associative ring with unity $1$ and consider $k\in \mathbb{N}$ such that $1+1+..+1=k$ is invertible. Denote by $\omega$ an arbitrary kth root of unity in $R$ and let $UT^{(k)}_{\infty}(R)$ be the group of upper triangular…

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs…

We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either…

In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of…

In this paper we count the number of k-potent elements over HZp , the quaternion algebra over Zp, and we give a descriptive formula for the general case. For k in {3, 4, 5}, we give an explicit formula for these values. Moreover, as an…

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called…

Idempotent elements play a fundamental role in ring theory, as they encode significant information about the underlying algebraic structure. In this paper, we study idempotent matrices from two perspectives. First, we analyze the partially…

We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We…

Denote by P(K, k) the members of the field K which are sums of kth powers of field elements, by P+(K, k) the set of members of K which are sums of kth powers of totally positive elements of K. We are interested in deciding whether or not…

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and $T_n(\mathbb{K})$ be the set of $n\times n$ lower triangular matrices with entries in $\mathbb{K}$. We show that $T_n(\mathbb{K})$ has dense subsemigroups that are generated by $n+1$…

In this paper, by using matix representation for quaternions andoctonions, we provide a procedure to obtain some example of k potent matrices of order 4 or 8, over the real field or over the field ZP, with p a prime number.

In this paper, with the help of trinomial coefficients we study some arithmetic properties of certain determiants involving reciprocals of binary quadratic forms over finite fields.

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict $k$-Hessenberg matrices and banded matrices.…

Let $n \geq 2$ be an integer. An \emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex…

We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\neq 2$ and for every positive integer $n$ such that $q^n>(k-1)^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of…

For a perfect field $k$, we construct a triangulated category of mixed motives over $k[t]/{(t^{m+1})}$. The ext groups in this category are given by higher Chow groups, and additive higher Chow groups.