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Register a new userReduction of matrices over simple Ore domains
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We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout domains.
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Let $R$ be a Bezout domain, and let $A,B,Cin R^{ntimes n}$ with $ABA=ACA$. If $AB$ and $CA$ are group invertible, we prove that $AB$ is similar to $CA$. Moreover, we have $(AB)^{#}$ is similar to $(CA)^{#}$. This generalize the main result of Cao and Li(Group inverses for matrices over a Bezout domain, {it Electronic J. Linear Algebra}, {bf 18}(2009), 600--612).
In this paper, we continue our study of the tensor product structure of category $mathcal W$ of weight modules over the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $chi$ and $chi(a)$ are different. Then we describe the Green ring $r(mathcal W)$ of the tensor category $mathcal W$. It is shown that $r(mathcal W)$ is isomorphic to the polynomial algebra over the group ring $mathbb{Z}hat{G}$ in one variable when $|chi(a)|=|chi|=infty$, and that $r(mathcal W)$ is isomorphic to the quotient ring of the polynomial algebra over the group ring $mathbb{Z}hat{G}$ in two variables modulo a principle ideal when $|chi(a)|<|chi|=infty$. When $|chi(a)|le|chi|<infty$, $r(mathcal W)$ is isomorphic to the quotient ring of a skew group ring $mathbb{Z}[X]sharphat{G}$ modulo some ideal, where $mathbb{Z}[X]$ is a polynomial algebra over $mathbb{Z}$ in infinitely many variables.
In this paper, we study the representations of the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(chi^{-1}, a, 0)$-modules under the assumption $|chi|=infty$ and $|chi|=|chi(a)|<infty$ respectively, and all finite dimensional indecomposable $kG(chi^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|chi|=|chi(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(chi^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra $kD_n$, where $n=2m$, $m>1$ odd, and char$(k)$=0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.
Let $R$ be a commutative additively idempotent semiring. In this paper, some properties and characterizations for permanents of matrices over $R$ are established, and several inequalities for permanents are given. Also, the adjiont matrices of matriecs over $R$ are considered. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices.
In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=mathbb{Z}/nmathbb{Z}$ and give some results that compute the value of this sum if $R$ is an arbitrary finite commutative ring $R$ for many values of $k$ and $d$. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ is always $0$ unless $d=2$, $textrm{card}(R) equiv 2 pmod 4$, $1<kequiv -1,0,1 pmod 6$ and the only element $ein R setminus {0}$ such that $2e =0$ is idempotent, in which case the sum is $textrm{diag}(e,e)$.
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