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• W2058194015 abstract "The main theorem proved in the present paper states as follows “Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m$><mml:mi>m</mml:mi></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$k$><mml:mi>k</mml:mi></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$n$><mml:mi>n</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$s$><mml:mi>s</mml:mi></mml:math>be fixed non-negative integers such that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$k$><mml:mi>k</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$n$><mml:mi>n</mml:mi></mml:math>are not simultaneously equal to<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$1$><mml:mn>1</mml:mn></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>be a left (resp right)<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$s$><mml:mi>s</mml:mi></mml:math>-unital ring satisfying<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left[ {left( {x^m y^k } right)^n - x^s y,x} right] = 0$><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>(resp<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left[ {left( {x^m y^k } right)^n - yx^s ,x} right] = 0$><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:mi>y</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>) Then<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>is commutative.” Further commutativity of left<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$s$><mml:mi>s</mml:mi></mml:math>-unital rings satisfying the condition<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$x^t left[ {x^m ,y} right] - y^r left[ {x,fleft( y right)} right]x^s = 0$><mml:msup><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$fleft( t right) in t^2 Zleft[ t right]$><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>Z</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m > 0,t,r$><mml:mi>m</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$s$><mml:mi>s</mml:mi></mml:math>are fixed non-negative integers, has been investigated Finally, we 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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$x$><mml:mi>x</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$y$><mml:mi>y</mml:mi></mml:math>for their values. These results generalize a number of commutativity theorems established recently." @default.
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• W2058194015 title "Commutativity of one sideds-unital rings through a Streb's result" @default.
• W2058194015 doi "https://doi.org/10.1155/s0161171297000367" @default.
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