In this paper we prove that on a 2-torsion free semiprime ring R every Jordan triple (;)-derivation (resp. generalized Jordan triple (;)derivation) on Lie ideal L is an (;)-derivation on L (resp. generalized (;)derivation on L)

Let [Formula: see text] be a semiprime ring, [Formula: see text] a square-closed Lie ideal of [Formula: see text] and [Formula: see text] a symmetric bi-derivation and [Formula: see text] be the trace of [Formula: see text] In this paper, we shall prove that [Formula: see text] contains a nonzero central ideal if any one of the following holds: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] (v) [Formula: see text] (vi) [Formula: see text] (vii) [Formula: see text] (viii) [Formula: see text] (ix) [Formula: see text] (x) [Formula: see text] (xi) [Formula: see text] and (xii) [Formula: see text], for all [Formula: see text]

In this work, the subject of ideal in a semiprime ring with multiplicative (generalized)- reverse derivations studied is included. We give new essential results for researchers in this field and generalize some results found in the literature. Also, the application of continuous reverse derivations in Banach algebras is discussed for the first time.

Let R be a semiprime ring, U a square-closed Lie ideal of R and D : R R ! R a symmetric reverse bi-derivation and d be the trace of D: In the present paper, we shall prove that R commutative ring if any one of the following holds: i) d(U) = (0); ii)d(U) Z; iii)[d (x) ; y] 2 Z; iv)d(x)oy 2 Z; v)d ([x; y])[d(x); y] 2 Z; vi)d (x y)(d(x)y) 2 Z; vii)d ([x; y])d(x)y 2 Z viii)d (x y) [d(x); y] 2 Z; ix)d(x) y [d(y); x] 2 Z; x)d([x; y]) (d(x) y) [d(y); x] 2 Z xi)[d(x); y] [g(y); x] 2 Z; for all x; y 2 U; where G : R R ! R is symmetric reverse bi-derivations such that g is the trace of

This study develops some results involving generalized homoderivation in semiprime rings and investigates the commutativity of semiprime rings admitting generalized homoderivations of ring R satisfying certain identities and some related results have also been discussed.