Generalized reverse derivations and commutativity of prime rings

Shuliang Huang

Communications in Mathematics (2019)

  • Volume: 27, Issue: 1, page 43-50
  • ISSN: 1804-1388

Abstract

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Let R be a prime ring with center Z ( R ) and I a nonzero right ideal of R . Suppose that R admits a generalized reverse derivation ( F , d ) such that d ( Z ( R ) ) ≠ 0 . In the present paper, we shall prove that if one of the following conditions holds: (i) F ( x y ) ± x y ∈ Z ( R ) , (ii) F ( [ x , y ] ) ± [ F ( x ) , y ] ∈ Z ( R ) , (iii) F ( [ x , y ] ) ± [ F ( x ) , F ( y ) ] ∈ Z ( R ) , (iv) F ( x ∘ y ) ± F ( x ) ∘ F ( y ) ∈ Z ( R ) , (v) [ F ( x ) , y ] ± [ x , F ( y ) ] ∈ Z ( R ) , (vi) F ( x ) ∘ y ± x ∘ F ( y ) ∈ Z ( R ) for all x , y ∈ I , then R is commutative.

How to cite

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Huang, Shuliang. "Generalized reverse derivations and commutativity of prime rings." Communications in Mathematics 27.1 (2019): 43-50. <http://eudml.org/doc/294158>.

@article{Huang2019,
abstract = {Let $R$ be a prime ring with center $Z(R)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d(Z(R))\ne 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F(xy)\pm xy\in Z(R)$, (ii) $F([x,y])\pm [F(x),y]\in Z(R)$, (iii) $F([x,y])\pm [F(x),F(y)]\in Z(R)$, (iv) $F(x\circ y)\pm F(x)\circ F(y)\in Z(R)$, (v) $[F(x),y]\pm [x,F(y)]\in Z(R)$, (vi) $F(x)\circ y\pm x\circ F(y)\in Z(R)$ for all $x,y \in I$, then $R$ is commutative.},
author = {Huang, Shuliang},
journal = {Communications in Mathematics},
keywords = {Prime rings; reverse derivations; generalized reverse derivations},
language = {eng},
number = {1},
pages = {43-50},
publisher = {University of Ostrava},
title = {Generalized reverse derivations and commutativity of prime rings},
url = {http://eudml.org/doc/294158},
volume = {27},
year = {2019},
}

TY - JOUR
AU - Huang, Shuliang
TI - Generalized reverse derivations and commutativity of prime rings
JO - Communications in Mathematics
PY - 2019
PB - University of Ostrava
VL - 27
IS - 1
SP - 43
EP - 50
AB - Let $R$ be a prime ring with center $Z(R)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d(Z(R))\ne 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F(xy)\pm xy\in Z(R)$, (ii) $F([x,y])\pm [F(x),y]\in Z(R)$, (iii) $F([x,y])\pm [F(x),F(y)]\in Z(R)$, (iv) $F(x\circ y)\pm F(x)\circ F(y)\in Z(R)$, (v) $[F(x),y]\pm [x,F(y)]\in Z(R)$, (vi) $F(x)\circ y\pm x\circ F(y)\in Z(R)$ for all $x,y \in I$, then $R$ is commutative.
LA - eng
KW - Prime rings; reverse derivations; generalized reverse derivations
UR - http://eudml.org/doc/294158
ER -

References

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