Multiplicative (generalized)-derivations and left ideals in semiprime rings

Year 2015, Volume: 44 Issue: 6, 1293 - 1306, 01.12.2015

Asma Ali , Basudeb Dhara , Shahoor Khan

Abstract

Let R be a semiprime ring with center Z(R). A mapping F : R → R
(not necessarily additive) is said to be a multiplicative (generalized)-
derivation if there exists a map f : R → R (not necessarily a derivation
nor an additive map) such that F (xy) = F (x)y + xf (y) holds for all
x,y ∈ R. The objective of the present paper is to study the following
identities: (i) F(x)F(y) ± [x,y] ∈ Z(R), (ii) F(x)F(y) ± x ◦ y ∈ Z(R),
(iii) F([x,y]) ± [x,y] ∈ Z(R), (iv) F(x ◦ y) ± (x ◦ y) ∈ Z(R), (v)
F([x,y]) ± [F(x),y] ∈ Z(R), (vi) F(x ◦ y) ± (F(x) ◦ y) ∈ Z(R), (vii)
[F(x),y] ± [G(y),x] ∈ Z(R), (viii) F([x,y]) ± [F(x),F(y)] = 0, (ix)
F(x◦y)±(F(x)◦F(y)) = 0, (x) F(xy)±[x,y] ∈ Z(R) and (xi)
F(xy)±x◦y ∈ Z(R) for all x,y in some appropriate subset of R, where
G : R → R is a multiplicative (generalized)-derivation associated with
the map g : R → R. 

Keywords

Semiprime ring, left ideal, derivation, multiplicative derivation, generalized derivation, multiplicative (generalized)-derivation

References

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