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A Basis for Z-Graded Identities of Matrices over Infinite Fields

Azevedo, Sergio (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 16R10, 16R20, 16R50The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z ), where α is the degree of the corresponding variable. This is a generalization of a result of Vasilovsky about the Z-graded identities...

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

Let m > 1 , s 1 be fixed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p ( x ) 0 , q = q ( x ) 0 , n = n ( x ) 0 , r = r ( x ) 0 such that either x p [ x n , y ] x q = x r [ x , y m ] y s or x p [ x n , y ] x q = y s [ x , y m ] x r for all y R . In the present paper it is shown that R is commutative if it satisfies the property Q ( m ) (i.e. for all x , y R , m [ x , y ] = 0 implies [ x , y ] = 0 ).

An identity related to centralizers in semiprime rings

Joso Vukman (1999)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to prove the following result: Let R be a 2 -torsion free semiprime ring and let T : R R be an additive mapping, such that 2 T ( x 2 ) = T ( x ) x + x T ( x ) holds for all x R . In this case T is left and right centralizer.

An identity with generalized derivations on Lie ideals, right ideals and Banach algebras

Vincenzo de Filippis, Giovanni Scudo, Mohammad S. Tammam El-Sayiad (2012)

Czechoslovak Mathematical Journal

Let R be a prime ring of characteristic different from 2 , U the Utumi quotient ring of R , C = Z ( U ) the extended centroid of R , L a non-central Lie ideal of R , F a non-zero generalized derivation of R . Suppose that [ F ( u ) , u ] F ( u ) = 0 for all u L , then one of the following holds: (1) there exists α C such that F ( x ) = α x for all x R ; (2) R satisfies the standard identity s 4 and there exist a U and α C such that F ( x ) = a x + x a + α x for all x R . We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of...

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.

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) ' ...

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