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Centers in domains with quadratic growth

Agata Smoktunowicz (2005)

Open Mathematics

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

Centralizers on semiprime rings

Joso Vukman (2001)

Commentationes Mathematicae Universitatis Carolinae

The main result: Let R be a 2 -torsion free semiprime ring and let T : R R be an additive mapping. Suppose that T ( x y x ) = x T ( y ) x holds for all x , y R . In this case T is a centralizer.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

Let be a commutative ring, 𝒢 be a generalized matrix algebra over with weakly loyal bimodule and 𝒵 ( 𝒢 ) be the center of 𝒢 . Suppose that 𝔮 : 𝒢 × 𝒢 𝒢 is an -bilinear mapping and that 𝔗 𝔮 : 𝒢 𝒢 is a trace of 𝔮 . The aim of this article is to describe the form of 𝔗 𝔮 satisfying the centralizing condition [ 𝔗 𝔮 ( x ) , x ] 𝒵 ( 𝒢 ) (and commuting condition [ 𝔗 𝔮 ( x ) , x ] = 0 ) for all x 𝒢 . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) 𝔗 𝔮 has the so-called proper form from a new perspective. Using the aforementioned...

Certain simple maximal subfields in division rings

Mehdi Aaghabali, Mai Hoang Bien (2019)

Czechoslovak Mathematical Journal

Let D be a division ring finite dimensional over its center F . The goal of this paper is to prove that for any positive integer n there exists a D ( n ) , the n th multiplicative derived subgroup such that F ( a ) is a maximal subfield of D . We also show that a single depth- n iterated additive commutator would generate a maximal subfield of D .

Classification of rings satisfying some constraints on subsets

Moharram A. Khan (2007)

Archivum Mathematicum

Let R be an associative ring with identity 1 and J ( R ) the Jacobson radical of R . Suppose that m 1 is a fixed positive integer and R an m -torsion-free ring with 1 . In the present paper, it is shown that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y R J ( R ) and (ii) [ x , [ x , y m ] ] = 0 , for all x , y R J ( R ) . This result is also valid if (ii) is replaced by (ii)’ [ ( y x ) m x m - x m ( x y ) m , x ] = 0 , for all x , y R N ( R ) . Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

Commutativity of rings through a Streb’s result

Moharram A. Khan (2000)

Czechoslovak Mathematical Journal

In this paper we investigate commutativity of rings with unity satisfying any one of the properties: { 1 - g ( y x m ) } [ y x m - x r f ( y x m ) x s , x ] { 1 - h ( y x m ) } = 0 , { 1 - g ( y x m ) } [ x m y - x r f ( y x m ) x s , x ] { 1 - h ( y x m ) } = 0 , y t [ x , y n ] = g ( x ) [ f ( x ) , y ] h ( x ) a n d [ x , y n ] y t = g ( x ) [ f ( x ) , y ] h ( x ) for some f ( X ) in X 2 [ X ] and g ( X ) , h ( X ) in [ X ] , where m 0 , r 0 , s 0 , n > 0 , t > 0 are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements x and y for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize...

Commutativity of rings with constraints involving a subset

Moharram A. Khan (2003)

Czechoslovak Mathematical Journal

Suppose that R is an associative ring with identity 1 , J ( R ) the Jacobson radical of R , and N ( R ) the set of nilpotent elements of R . Let m 1 be a fixed positive integer and R an m -torsion-free ring with identity 1 . The main result of the present paper asserts that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y R J ( R ) and (ii) [ ( x y ) m + y m x m , x ] = 0 = [ ( y x ) m + x m y m , x ] , for all x , y R J ( R ) . This result is also valid if (i) and (ii) are replaced by (i) ' ...

Commutativity of rings with polynomial constraints

Moharram A. Khan (2002)

Czechoslovak Mathematical Journal

Let p , q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s -unital ring satisfying [ f ( x p y q ) - x r y , x ] = 0 ( [ f ( x p y q ) - y x r , x ] = 0 , respectively) where f ( λ ) λ 2 [ λ ] , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.

Commutativity theorems for rings with differential identities on Jordan ideals

L. Oukhtite, A. Mamouni, Mohammad Ashraf (2013)

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate commutativity of ring R with involution ' * ' which admits a derivation satisfying certain algebraic identities on Jordan ideals of R . Some related results for prime rings are also discussed. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

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