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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...

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