Some results on prime rings with multiplicative derivations

Abstract

Let R be a prime ring with center Z(R) and an automorphism α. A mapping δ : R → R is calledmultiplicative skew derivation if δ(xy) = δ(x)y + α(x)δ(y) for all x, y ∈ R and a mapping F : R → R is saidto be multiplicative (generalized)-skew derivation if there exists a unique multiplicative skew derivation δ such that F(xy) = F(x)y + α(x)δ(y) for all x, y ∈ R. In this paper, our intent is to examine the commutativity of R involving multiplicative (generalized)-skew derivations that satisfy the following conditions: $(i);F(mathcal x^2);+;delta(mathcal x);=;delta(mathcal x^2;);+;mathcal xF(mathcal x),;(ii);F(mathcal x;◦;mathcal y);=;delta(mathcal x;◦;mathcal y);pm;mathcal x;◦;mathcal y;,;(iii);F(lbrackmathcal x,;mathcal yrbrack);=;delta(lbrackmathcal x,;mathcal y);pm;lbrackmathcal x,;mathcal yrbrack,;(iv);F(mathcal x^2;);=;delta(mathcal x^2;),;(v);F(lbrackmathcal x,;mathcal yrbrack);=;pm x;k;lbrack x,;delta(y)rbrack x;m;,;(vi);F(x;◦;y);=;pmmathcal x^k;(mathcal x;◦;delta(mathcal y))mathcal x^m,;(vii);F(lbrackmathcal x,;mathcal yrbrack);=;pmmathcal x^klbrackdelta(mathcal x),;mathcal yrbrackmathcal x^m;,;(viii);F(mathcal x;◦;mathcal y);=;pmmathcal x(delta(mathcal x);◦;mathcal y)mathcal x^m;for;all;mathcal x,;mathcal y;in;R.$