scholarly journals Decomposition of Automorphisms of Certain Solvable Subalgebra of Symplectic Lie Algebra over Commutative Rings

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Xing Tao Wang ◽  
Lei Zhang
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Explicit Description ◽  
Upper Triangular Matrix

LetCl+1(R)be the2(l+1)×2(l+1)matrix symplectic Lie algebra over a commutative ringRwith 2 invertible. Thentl+1CR  =  {m-1m-20-m-1T ∣ m̅1is anl+1upper triangular matrix,m̅2T=m̅2,  over  R}is the solvable subalgebra ofCl+1(R). In this paper, we give an explicit description of the automorphism group oftl+1(C)(R).

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

Orthogonal decompositions of Lie algebras over finite commutative rings

2021 ◽  
pp. 2350006
Author(s):  
Songpon Sriwongsa
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Orthogonal Decomposition ◽  
Necessary Condition ◽  
Special Linear ◽  
Orthogonal Lie Algebra ◽  
Orthogonal Decompositions ◽  
Finite Commutative Ring

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

scholarly journals DECOMPOSITION OF JORDAN AUTOMORPHISMS OF TRIANGULAR MATRIX ALGEBRA OVER COMMUTATIVE RINGS

2010 ◽  
Vol 52 (3) ◽  
pp. 529-536 ◽  
Author(s):  
XING TAO WANG ◽  
YUAN MIN LI
Keyword(s):  
Commutative Ring ◽  
Matrix Algebra ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Graph Automorphism ◽  
Triangular Matrix Algebra ◽  
Inner Automorphism

AbstractLet Tn+1(R) be the algebra of all upper triangular n+1 by n+1 matrices over a 2-torsionfree commutative ring R with identity. In this paper, we give a complete description of the Jordan automorphisms of Tn+1(R), proving that every Jordan automorphism of Tn+1(R) can be written in a unique way as a product of a graph automorphism, an inner automorphism and a diagonal automorphism for n ≥ 1.

ON RINGS DETERMINED BY ZERO PRODUCTS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350058 ◽  
Author(s):  
HOGER GHAHRAMANI
Keyword(s):  
Operator Algebras ◽  
Triangular Matrix ◽  
Additive Group ◽  
Commutative Rings ◽  
Matrix Rings ◽  
Full Matrix ◽  
Additive Map ◽  
Upper Triangular Matrix

Let [Formula: see text] be a ring. We say that [Formula: see text] is zero product determined if for every additive group [Formula: see text] and every bi-additive map [Formula: see text] the following holds: if ϕ(a, b) = 0 whenever ab = 0, then there exists an additive map [Formula: see text] such that ϕ(a, b) = T(ab) for all [Formula: see text]. In this paper, first we study the properties of zero product determined rings and show that semi-commutative and non-commutative rings are not zero product determined. Then, we will examine whether the rings with a nontrivial idempotent are zero product determined. As applications of the above results, we prove that simple rings with a nontrivial idempotent, full matrix rings and some classes of operator algebras are zero product determined rings and discuss whether triangular rings and upper triangular matrix rings are zero product determined.

On Quasipolar Rings

2012 ◽  
Vol 19 (04) ◽  
pp. 683-692 ◽  
Author(s):  
Zhiling Ying ◽  
Jianlong Chen
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Ring Theory ◽  
Clean Rings ◽  
Commutative Local Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring ◽  
Regular Rings

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

scholarly journals m-commuting Additive Maps on Upper Triangular Matrices Rings

2019 ◽  
pp. 505-514
Author(s):  
Driss Aiat Hadj Ahmed
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Additive Map ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Additive Maps ◽  
Upper Triangular Matrix Ring

Let $T_{n}(R)$ be the upper triangular matrix ring over a unital commutative ring whose characteristic is not a divisor of $m$. Suppose that $f:T_{n}(R)\rightarrow T_{n}(R)$ is an additive map such that $X^{m}f(X)=f(X)X^{m}$ for all $x \in T_{n}(R),$ where $m\geq 1$ is an integer. We consider the problem of describing the form of the map $X \rightarrow f(X)$.

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