scholarly journals Certain Properties of Square Matrices over Fields with Applications to Rings

2021 ◽  
Vol 54 (2) ◽  
pp. 109-116
Author(s):  
Peter V. Danchev
Keyword(s):  
Linear Algebra ◽  
Present Author ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Matrix ◽  
Zero Matrix ◽  
Siberian Math ◽  
Diagonalizable Matrix ◽  
Regular Rings ◽  
Square Matrix

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.

scholarly journals On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

2021 ◽  
Vol 14 (5) ◽  
pp. 547-553
Author(s):  
Peter Danchev ◽  
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Closed Field ◽  
Multilinear Algebra ◽  
Infinite Field ◽  
Algebraically Closed Field ◽  
Periodic Matrix ◽  
Nilpotent Matrix ◽  
Square Matrix

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

Matrix Commutators

1961 ◽  
Vol 13 ◽  
pp. 353-355 ◽  
Author(s):  
M. F. Smiley
Keyword(s):  
Closed Field ◽  
Classical Theorem ◽  
Algebraically Closed Field ◽  
Square Matrix

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

Multiplication complexe et équivalence élémentaire dans le langage des corps (Complex multiplication and elementary equivalence in the language of fields)

2002 ◽  
Vol 67 (2) ◽  
pp. 635-648
Author(s):  
Xavier Vidaux
Keyword(s):  
Function Fields ◽  
Complex Multiplication ◽  
Closed Field ◽  
Constant Symbol ◽  
Endomorphism Rings ◽  
Modular Invariant ◽  
Elementary Equivalence ◽  
Algebraically Closed Field

AbstractLet K and K′ be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0. non k-isomorphic, and let C and C′ be two curves with respectively K and K′ as function fields. We prove that if the endomorphism rings of the curves are not isomorphic then K and K′ are not elementarily equivalent in the language of fields expanded with a constant symbol (the modular invariant). This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

Sign in / Sign up

Export Citation Format

Share Document