Displaying similar documents to “A commutativity theorem for associative rings”

Commutativity of rings through a Streb’s result

Moharram A. Khan (2000)

Czechoslovak Mathematical Journal

Similarity:

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...

Classification of rings satisfying some constraints on subsets

Moharram A. Khan (2007)

Archivum Mathematicum

Similarity:

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 with constraints involving a subset

Moharram A. Khan (2003)

Czechoslovak Mathematical Journal

Similarity:

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) ' [ x m , y m ] = 0 for all x , y R N ( 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 N ( R ) . Other similar...

On ( σ , τ ) -derivations in prime rings

Mohammad Ashraf, Nadeem-ur-Rehman (2002)

Archivum Mathematicum

Similarity:

Let R be a 2-torsion free prime ring and let σ , τ be automorphisms of R . For any x , y R , set [ x , y ] σ , τ = x σ ( y ) - τ ( y ) x . Suppose that d is a ( σ , τ ) -derivation defined on R . In the present paper it is shown that ( i ) if R satisfies [ d ( x ) , x ] σ , τ = 0 , then either d = 0 or R is commutative ( i i ) if I is a nonzero ideal of R such that [ d ( x ) , d ( y ) ] = 0 , for all x , y I , and d commutes with both σ and τ , then either d = 0 or R is commutative. ( i i i ) if I is a nonzero ideal of R such that d ( x y ) = d ( y x ) , for all x , y I , and d commutes with τ , then R is commutative. Finally a related result has been obtain...