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Displaying similar documents to “Commutativity of rings with constraints involving a subset”

Classification of rings satisfying some constraints on subsets

Moharram A. Khan (2007)

Archivum Mathematicum

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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

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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...

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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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 ).