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- W3108590908 abstract "Abstract We establish sufficient conditions for 3-prime near-rings to be commutative rings. In particular, for a 3-prime near-ring R with a derivation d , we investigate conditions such as $$d([U,V])subseteq Z(R)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mo>[</mml:mo> <mml:mi>U</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>]</mml:mo> <mml:mo>)</mml:mo> <mml:mo>⊆</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , $$d(U)subseteq Z(R)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>U</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⊆</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , $$x_{o}d(R)subseteq Z(R)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mi>d</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊆</mml:mo> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , and $$Ux_{o}subseteq Z(R)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>U</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mo>⊆</mml:mo> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . As a by-product, we generalize and extend known results related to rings and near-rings. Furthermore, we discuss the converse of a well-known result in rings and near-rings, namely: if $$xin Z(R)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , then $$d(x)in Z(R)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In addition, we provide useful examples illustrating our results." @default.
- W3108590908 created "2020-12-07" @default.
- W3108590908 creator A5013877839 @default.
- W3108590908 date "2020-11-22" @default.
- W3108590908 modified "2023-09-25" @default.
- W3108590908 title "On some properties of near-rings" @default.
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- W3108590908 doi "https://doi.org/10.1007/s40065-020-00302-0" @default.
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