Related papers: Idempotent factorization on some matrices over qua…

In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse…

We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting…

Let $X=(x_{ij})$ and $Y=(y_{ij})$ be generic $n$ by $n$ matrices and $Z=XY-YX$. Let $S=k[x_{11},...,x_{nn},y_{11},...,y_{nn}]$, where $k$ is a field, let $I$ be the ideal generated by the entries of $Z$ and let $R=S/I$. We give a conjecture…

This paper aims to continue the studies initiated by Botha in [Linear Algebra Appl. 273 (1998), 65-82; Linear Algebra Appl. 286 (1999), 37-44; Linear Algebra Appl. 315 (2000), 1-23] by extending them to matrices over noncommutative division…

We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the convolution of kernels defining integral functors between…

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is…

In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for…

We study the class of those linear relations that can be factorized as products of idempotent relations. We provide several characterizations of this class, extending known factorization results for operators to the more general setting of…

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that…

For an element $a$ of an integral domain D under an equivalence relation \tau, the \tau-factorization of a is defined as \lambda a_1 a_2... a_k, where \lambda is a unit in D and a_i \tau a_j for all i, j. An irreducible element has no…

In a Dedekind domain $D$, every non-zero proper ideal $A$ factors as a product $A=P_1^{t_1}\cdots P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain $D$, the $D$-modules $D/P_i^{t_i}$ are uniserial. We extend this…

We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to…

We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular…

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all…

A commutator of unipotent matrices of index 2 is a matrix of the form $XYX^{-1}Y^{-1}$, where $X$ and $Y$ are unipotent matrices of index 2, that is, $X\ne I_n$, $Y\ne I_n$, and $(X-I_n)^2=(Y-I_n)^2=0_n$. If $n>2$ and $\mathbb F$ is a field…

We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…

The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by…

I provide an alternate and self-contained proof of Botha's theorem on products of idempotent and square-zero matrices where the product contains two square-zero factors, and provide a conclusive characterisation of products of singular…

We show that for a multivariable polynomial $p(z)=p(z_1, \ldots , z_d)$ with a determinantal representation $$ p(z) = p(0) \det (I_n- K (\oplus_{j=1}^d z_j I_{n_j}))$$ the matrix $K$ is structurally similar to a strictly $J$-contractive…