Derivations and reverse derivations in semiprime rings

Abstract

The notion of reverse derivation is studied and some properties are obtained. It is shown that in the class of semiprime rings, this notion coincides with the usual derivation when it maps a semiprime ring into its center. However, we provide some examples to show that it is not the case in general. Also it is shown that non-commutative prime rings do not admit a non-trivial skew commuting derivation. Mathematics Subject Classification: 16A70, 16N60, 16W25Keywords: Prime ring, semiprime ring, anticommutative, derivation, reverse derivation, skew commuting map PreliminariesThroughout, R denotes a ring with center Z(R). We write [x, y] for xy − yx. Recall that a ring R is called prime if aRb = 0 implies a = 0 or b = 0; and

Ideal of Prime -Rings with Right Reverse Derivations

“…Recall that a -ring M is called prime if aMb = 0 implies a=0 or b=0and it is called semiprime if aMa = 0 implies a=0 ,a -ring M is called commutative if [ , ]  = 0 for all x,yM and ,Bresar and Vakman [2] have introduced the notion of a reverse derivation, the reverse derivation on semi prime rings have been studied by Samman and Alyamani [6] and K.KDey, A.IC.Paul, I.S.Rakhimov [3] have introduced the concepts of reverse derivation on -ring as an additive mapping d from M in to M is called reverse derivation if d(xy) = d(y)x + yd(x), for all x,yM ,and we consider an assumption (*) by xyz = xyz for all x,y,zU,,, where U is ideal of -ring.…”

Generalized reverse derivations and commutativity of prime rings

“…It is proved that if R is a prime ring and d a nonzero reverse derivation of R, then R is a commutative integral domain and d is just a derivation. The reverse derivation on semiprime rings have been studied in [15]. The concept of reverse derivation has relations with some generalizations of derivations.…”

Some Results on Symmetric Reverse ∗-n-Derivations

“…A square closed Lie ideal of such that ⊈ is called an admissible Lie ideal of [11]. Relationship between derivations and reverse derivations with examples were given by [13]. Recently there has been a great deal of work done by many authors on commuting and centralizing mappings on prime rings and semiprime rings, see ([4], [5], [6], [9], [10]).…”