## Some results on prime rings with multiplicative derivations

TURKISH JOURNAL OF MATHEMATICS, vol.44, no.4, pp.1401-1411, 2020 (Peer-Reviewed Journal)  • Publication Type: Article / Article
• Volume: 44 Issue: 4
• Publication Date: 2020
• Doi Number: 10.3906/mat-2002-24
• Journal Name: TURKISH JOURNAL OF MATHEMATICS
• Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
• Page Numbers: pp.1401-1411
• Keywords: Prime ring, multiplicative generalized derivation, multiplicative (generalized)-skew derivation, multiplicative left centralizer, (GENERALIZED)-DERIVATIONS, COMMUTATIVITY

#### Abstract

Let R be a prime ring with center Z(R) and an automorphism a. A mapping delta : R -> R is called multiplicative skew derivation if delta(xy) = delta(x)y + alpha(x)delta(y) for all x, y is an element of R and a mapping F : R -> R is said to be multiplicative (generalized)-skew derivation if there exists a unique multiplicative skew derivation delta such that F(xy) = F(x)y + alpha(x)delta(y) for all x, y is an element of 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(x(2)) + x delta(x) = delta(x(2)) + xF(x), (ii) F(x circle y) = delta(x circle y) +/- x circle y, (iii) F([x, y]) = delta([x, y])+/-[x, y], (iv) F(x(2)) = delta(x(2)), (v) F([x, y]) = +/- x(k) [x, delta(y)]x(m), (vi) F(x circle y) = +/- x(k)(x circle delta(y))x(m), (vii) F([x, y]) = +/- x(k) [delta(x), y]x(m), (viii) F(x circle y) = +/- x(delta(x) circle y)x(m) for all x, y is an element of R.