Semiprime rings with nilpotent Lie ring of inner derivations

. We give an elementary and self-contained proof of the theorem which says that for a semiprime ring commutativity, Lie-nilpotency, and nilpotency of the Lie ring of inner derivations are equivalent conditions.


Preliminaries and introduction
Throughout the text R stands for an associative ring (possibly without identity) and n for a positive integer. By Z(R) we denote the center of R. The ring R is said to be semiprime, if ∀ a ∈ R : aRa = {0} =⇒ a = 0.
Let us recall that R may be regarded as a Lie ring with the Lie multiplication defined by [x, y] = xy − yx. For n 3 and x 1 , . . . , x n ∈ R we define inductively The ring R is said to be Lie-nilpotent, if there exists an n such that [x 1 , . . . , x n+1 ] = 0 for all x 1 , . . . , x n+1 ∈ R. (Notice that R is Lie-nilpotent whenever it is commutative).
A map d: R → R is called a derivation, if it is additive and satisfies the Leibniz The set Der(R) of all derivations d: R → R is a Lie ring under the pointwise addition and the Lie multiplication defined by AMS (2010) Subject Classification: primary 16W10; secondary 16N60, 16W25.

[104]
Kamil Kular Notice that the Lie ring D is abelian if and only if it is nilpotent of class 1.
Let a ∈ R. It is easy to see that ∂ a : R x → [a, x] ∈ R is a derivation. This derivation is referred to as the inner derivation generated by a. One can prove that IDer(R) = {∂ a : a ∈ R} is a Lie subring of Der(R) (cf. Proposition 2.2 in the sequel).
Commutativity of semiprime rings with derivations has been studied by several authors. In [3] Daif and Bell proved that a semiprime ring R is commutative whenever it admits a derivation d such that for all x, y ∈ R. A generalization of the Daif and Bell result can be found in [5] (see also [2] where some related results are presented). Argaç and Inceboz proved that a semiprime ring R is commutative whenever there exist a derivation d: R → R and a positive integer n such that for all x, y ∈ R (see [1]).
The aim of the present note is to give an elementary and self-contained proof of Theorem 1.1 Suppose that R is semiprime. Then the following conditions are equivalent: (1) the Lie ring IDer(R) is nilpotent, Even though similar results are known, the theorem seems to be not explicitly stated in the literature. We will present a nice application of this theorem to the study of abstract derivations in the next paper.
We refer the reader to [4] for terminology, definitions and basic facts in ring theory.

Some lemmas and useful facts
Before presenting the elementary proof of Theorem 1.1, we collect some lemmas and useful remarks. The first lemma is very well known. We include its proof for the sake of completness. Proof. If a ∈ R, then Using the above lemma we immediately obtain The next two lemmas seem to be of separate interest.

Lemma 2.3
Let x ∈ R. The following conditions are equivalent:

e., ∂ x is a central element of the Lie ring
IDer(R)),

Proof. By Lemma 2.1, condition (1) is satisfied if and only if
∀ y, a ∈ R : ∂ ∂x(y) (a) = 0, and this means exactly that ∂ x (y) ∈ Z(R) for any y ∈ R. On the other hand, it is obvious that (1) Proof. Implication (1) ⇒ (2) is an immediate consequence of Proposition 2.2 (i) and the previous lemma. So let us assume that (2) is satisfied. Then for all x 1 , . . . , x n+1 ∈ R. [106]

Kamil Kular
The last lemma will also play a crucial role in the sequel.

Main results
Let us begin with a quite simple theorem.

Theorem 3.1
Suppose that R is semiprime and the Lie ring IDer(R) is abelian. Then R is commutative.
The following result is the technical heart of the note.  This proves implication (3) ⇒ (1).
Let us conclude by a natural example of a Lie-nilpotent ring which is not commutative.

Example
Consider a field F and the matrix ring