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Let R be a left artinian central F-algebra, T(R) = J(R) + [R,R], and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of R = R/J(R) is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson. © Inst. Math. CAS 2001  

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if RZD(R) is finite, then R is finite  

In the proof of Subcase 2 of Theorem 5 in Mahdavi-Hezavehi (2004) [2], not all the required steps are considered properly. Here we shall deal with the remaining step of Subcase 2. Therefore, this completes the proof of the main result that if D is an F-central finite-dimensional division algebra and M is a maximal subgroup of GLn (D), D ≠ F, n > 1, and M / M ∩ F* is torsion, then M is abelian-by-finite. © 2009 Elsevier Inc. All rights reserved  

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(Mn(F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GLn(F) and SLn(F). We show that Γ(Mn(F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GLn(F)) and Γ(SL n(F)). Also we show... 

Let R be a non-commutative ring. The commuting graph of R denoted by Λ (R), is a graph with vertex set R Z(R), and two distinct vertices a and b are adjacent if ab = ba. In this paper we investigate some properties of Λ(R), whenever R is a finite semisimple ring. For any finite field F, we obtain minimum degree, maximum degree and clique number of Λ(M n (F)). Also it is shown that for any two finite semisimple rings R and S, if Λ(R) ≃ Λ(S), then there are commutative semisimple rings R1 and S1 and semisimple ring T such that R ≃T × R1, S ≃ T × S1 and |R1| = |S1|. © 2004 Elsevier Inc. All rights reserved  

The zero-divisor graph of a ring R is defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if x y = 0. Recently, it has been shown that for any finite ring R, Γ (R) has an even number of edges. Here we give a simple proof for this result. In this paper we investigate some properties of zero-divisor graphs of matrix rings and group rings. Among other results, we prove that for any two finite commutative rings R, S with identity and n, m ≥ 2, if Γ (Mn (R)) ≃ Γ (Mm (S)), then n = m, | R | = | S |, and Γ (R) ≃ Γ (S). © 2007 Elsevier Inc. All rights reserved  

In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if xy = 0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ (R). In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Γ (R) ≃ Γ (S), then R ≃ S. For any finite field F and each integer n ≥ 2, we prove that if R is a ring and Γ (R) ≃ Γ (Mn), then R ≃ Mnn. Redmond defined the simple... 

Recently, it is shown that, if D is a finite-dimensional division ring, then GLn(D) is not finitely generated. Our object here is to provide a general framework for the groups of units of left Artinian rings. We prove that, if R is an infinite F-algebra of finite dimension over F, then U(R) is not finitely generated. We show that any infinite subnormal subgroup of GL n (D) has no finite maximal subgroup. Also, we prove that for any infinite left Artinian ring R, U(R) has no finite maximal subgroup, which is a result analogous to that for rings. © Inst. Math. CAS 2002  

In order to design Button Beam Position Monitors (BPMs) for synchrotron facilities, one algorithm by C# have been developed which can calculate all required parameters to analyze optimal design based on vacuum chamber and button dimensions. Beam position monitors are required to get beam stabilities on submicron levels. For this purpose, different parameters such as capacitance, sensitivity versus bandwidth, intrinsic resolution, induced charge and voltage on buttons are calculated. Less intrinsic resolution and high sensitivity and capacitance are desired. To calculate induced charge and voltage on each button, Poisson's equation has been solved by Green method. For sensitivities... 

Let D be a division algebra algebraic over its center F. Given a (Krull) valuation v on F, it is shown that v extends to a valuation on D if and only if for each separable element c∈D′ there exists a valuation w on K: = F(c) extending v on F such that (Formula presented.), where D′ is the derived group of D* and W* is the unit group of the valuation ring W of w. © 2017 Taylor & Francis  

Let D be a division algebra algebraic over its center F. Given a (Krull) valuation v on F, it is shown that v extends to a valuation on D if and only if for each separable element c∈D′ there exists a valuation w on K: = F(c) extending v on F such that K ∩ D’ ⊂ W*,, where D′ is the derived group of D* and W* is the unit group of the valuation ring W of w. © 2017 Taylor & Francis  

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n ≥ 1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D : F] < ∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M = K* where K is a field and [D : F] < ∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is... 

Let D be an infinite division algebra of finite dimension over its centre Z(D) = F, and n a positive integer. The structure of maximal subgroups of skew linear groups are investigated. In particular, assume N is a normal subgroup of GLn(D) and M is a maximal subgroup of N containing Z(N). It is shown that if M/Z(N) is finite, then N is central. © Inst. Math. CAS 2002  

Let D be an infinite division algebra of finite dimension over its center. Assume that N is a subnormal subgroup of GLn(D) with n≥1. It is shown that if N is finitely generated, then N is central. © 2000 Academic Press  

The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal  

Let D be a division ring with center F. An element of the form xyx -1y-1 ∈ D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In this paper it is shown that if D is algebraic over F and Char(D) = 0, then D = T(D). We conjecture that it is true in general. Among other results it is shown that in characteristic zero if T(D) is algebraic over F, then D is algebraic over F  

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd D/F(D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK 1(D) = 1 and F *2 = F *2n. Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated  

Given a division ring D with center F, the structure of maximal subgroups M of GL n(D) is investigated. Suppose D ≠ F or n > 1. It is shown that if M/(M ∩ F*) is locally finite, then char F = p > 0 and either n = 1, [D:F] = p 2 and M ∪ {0} is a maximal subfield of D, or D = F, n = p, and M ∪ {0} is a maximal subfield of M p(F), or D = F and F is locally finite. It is also proved that the same conclusion holds if M/(M ∩ F*) is torsion and D is of finite dimension over F. Furthermore, it is shown that if the r-th derived group M (r) of M is locally finite, then either M (r) is abelian or F is algebraic over its prime subfield  

Let D be an F-central non-commutative division ring. Here, it is proved that if GL n(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GL n(F) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL 2(F) for fields F with Char F ≠ 2  

This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for functors on Artin local rings. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic applications