THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2013.
Oct,
20(4):
243-249

- Received : April 02, 2013
- Accepted : August 29, 2013
- Published : October 30, 2013

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Let
A
be an algebra and
D
a derivation of
A
. Then
D
is called algebraic
nil
if for any
x
∈
A
there is a positive integer
n
=
n
(
x
) such that
D
^{n(x)}
(
P
(
x
)) = 0; for all
P
∈ ℂ (by convention
D
^{n(x)}
(α) = 0; for all α ∈ ℂ). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra
A
maps into
N
(
A
) ; where
N
(
A
) denotes the set of all nilpotent elements of
A
. As an application, we deduce that any nilpotent derivation on a commutative complex algebra
A
maps into
N
(
A
).
Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.
A
be a complex algebra. A linear map
D
from
A
to
A
is called a
derivation
if
D
(
xy
) =
D
(
x
)
y
+
xD
(
y
) holds for all
x, y
∈
A
. A derivation of
D
on
A
is called
nil
if for any
x
∈
A
there is a positive integer
n
=
n
(
x
) such that
D
^{n(x)}
= 0 (see
[6]
). Here, if the number
n
can be taken independently of
x
,
D
is called
nilpotent
. A derivation
D
of
A
is called
algebraic nil
if for any
x
∈
A
there is a positive integer
n
=
n
(
x
) such that
D
^{n(x)}
(
P
(
x
)) = 0, for all
P
∈ ℂ[
X
] (by convention
D
^{n(x)}
(α) = 0, for all α ∈ ℂ).
We will denote by
Q
(
A
) the set of all quasinilpotent elements in a Banach algebra
A
. In 1955, Singer and Wermer
[12]
proved that a continuous derivation on a commutative Banach algebra maps into the (Jacobson) radical, and they conjectured that this result holds even if the derivation is discontinuous. In 1988, Thomas
[13]
solved the long standing problem by showing that the conjecture is true.
In 1991, Kim and Jun
[10]
proved that if
D
is a derivation on a noncommutative Banach algebra
A
satisfying the condition [[
A, A
],
A
] = 0 then
D
(
A
) ⊂
Q
(
A
). In 1992, Vukman
[15]
proved that if
D
is a linear Jordan derivation on a noncommutative Banach algebra
A
such that the map
F
(
x
) = [[
Dx
,
x
],
x
] is commuting on
A
then
D
= 0. In 1992, Mathieu and Runde
[11]
proved that if
D
is a centralizing derivation on a Banach algebra A; then
D
(
A
) ⊂
rad
(
A
): In 1994, Bresar
[5]
showed that if
D
is a bounded derivation of a Banach algebra such that [
D
(
x
),
x
] ∈
Q
(
A
) for every
x
∈
A
; then
D
(
A
) ⊂
rad
(
A
) where rad(A) denotes the Jacobson radical of
A
.
To the best of our knowledge, there is no inclusion versions for derivations on arbitrary algebra, except the paper of Colville, Davis, and Keimel
[9]
in which they began studying positive derivations on f-rings (i.e.,
D
(
a
) ≥ 0, for all
a
≥ 0) and the papers of Boulabiar
[4]
, A. Toumi et al
[14]
and Ben Amor
[2]
, in which the authors studied exclusively positive and order bounded derivations on Archimedean almost
f
-algebras.
It is well-known that the notion of nil derivations is a generalization of the notion of nilpotent derivations. The latter, because of its close relation with automorphisms and the existence of a Jordan decomposition into semisimple and nilpotent parts for a large family of derivations (it is a generalization of that of algebraic derivations), has received considerable attention (see
[6
,
7
,
8]
). In this paper we shall be concerned principally with the range of algebraic nil derivations
D
on commutative algebra, on noncommutative archimedean d-algebra and on noncommutative algebra
A
satisfying the following condition; [[
A
,
A
],
A
] = 0.
Proposition 2.1.
Let A be a commutative complex algebra, n be a positive integer, D be a derivation on A and x ∈ A such that
D^{n}
(
x
),
D^{n}
(
x
^{2}
),
D^{n}
(
x^{n}
) ∈
N
(
A
),
where N
(
A
)
denotes the set of all nilpotent elements of A
.
Then
D
(
x
) ∈
N
(
A
).
Proof
. Let
x
∈ A with
D^{n}
(
x
),
D^{n}
(
x
^{2}
),
D^{n}
(
x^{n}
) ∈
N
(
A
). It follows that
Since
D^{n}
(
x
) ∈
N
(
A
) ; we have
Moreover, letting
such that
n_{1}
+
n_{2}
=
n
, this one has
By using the Leibnitz rule for
D^{k}
(
x^{n1}
) and
D^{k}
(
x^{n2}
) in Equality (3) and by using the relation (2), we deduce that
(
D
(
x
))
^{n}
∈
N
(
A
)
and then
D
(
x
) ∈
N
(
A
).□
From the above result, we deduce the following:
Proposition 2.2.
Let A be a commutative complex algebra, n be a positive integer, D be a derivation on A and x ∈ A such that
D^{n}
(
x
) =
D^{n}
(
x
^{2}
) =
D^{n}
(
x^{n}
) = 0.
Then (
D
(
x
))
^{n}
= 0.
The below theorem is an immediate consequence of Proposition 2.2.
Theorem 2.3.
Let A be a commutative complex algebra and let D be an algebraic nil derivation on A. Then D
(
A
)
is contained in N
(
A
).
Since any nilpotent derivation is algebraic nil, we have the following:
Corollary 2.4.
Let A be a commutative complex algebra and let D be a nilpotent derivation on A. Then D
(
A
)
is contained in N
(
A
).
In what follows, we shall deal with the range of algebraic nil derivation on non-commutative algebras. In order to hit this mark, we will need the following lemma.
Lemma 2.5
([10, Lemma 3.1]).
Let A be a complex algebra satisfying the condition
[[
A, A
],
A
] = 0.
Let A
⊕
A be the vector space direct sum. Define a multiplication in A
⊕
A by setting
(
a_{1}, b_{1}
) (
a_{2}, b_{2}
) = (
a_{1}a_{2}
+
a_{2}a_{1}, b_{1}b_{2}
+
b_{2}b_{1}
)
for all
(
a_{1}, b_{1}
), (
a_{2}, b_{2}
)
in A
⊕
A
.
Then A
⊕
A is a commutative algebra
.
Using the previous lemma, we deduce the following result. Its proof is inspired from [10, Theorem 3.2].
Theorem 2.6.
Let A be a complex algebra satisfying the condition
[[
A, A
],
A
] = 0
and let D be an algebraic nil derivation on A. Then D
(A)
is contained in N
(
A
).
Proof
. By the previous lemma,
A
⊕
A
is a commutative algebra. Now we define the linear mapping
:
A
⊕
A
→
A
⊕
A
by
(
a, b
) = (
D
(
a
),
D
(
b
)).
Since
D
is an algebraic nil derivation on
A
, it is not hard to prove that
is an algebraic nil derivation on
A
⊕
A
. By Theorem 1, we have
(
A
⊕
A
) ⊂
N
(
A
⊕
A
) =
N
(
A
) ⊕
N
(
A
). Therefore
D
(
A
) ⊂
N
(
A
).□
Corollary 2.7.
Let A be a complex algebra satisfying the condition
[[
A
,
A
],
A
] = 0
and let D be a nilpotent derivation on A. Then D
(
A
)
is contained in N
(
A
).
Next, we will be interested with the range of derivations on noncommutative algebra
A
satisfying the following condition;
(χ)
a
[
A, A
]
b
= 0
for all
a,b
∈
A
.
Theorem 2.8.
Let A be a complex algebra satisfying the condition (χ) and let D be an algebraic nil derivation on A. Then D(A) is contained in N
(
A
).
Proof
. Let
x
∈
A
. Then there exists
such that
D^{n}
(
x
) =
D^{n}
(
x
^{2}
) =
D^{n}
(
x^{n}
) = 0: Let
a, b
∈
A
. It follows that
Moreover, let
n
_{1}
;
n
_{2}
∈ℕ N such that
n
_{1}
+
n
_{2}
=
n
(
x
), then
By using the Leibnitz rule for
aD^{k}
(
x^{n1}
)
b
and
aD^{k}
(
x^{n2}
)
b
in Equality (5), by using Equality (4) and taking into account that
D^{n}
(
x
) = 0, we deduce that
a
(
D
(
x
))
^{n} b
= 0
for all
a, b
∈
A
. Consequently (
D
(
x
))
^{n+2}
= 0. Therefore
D
(
A
) ⊂
N
(
A
).□
Corollary 2.9.
Let A be a complex algebra satisfying the condition
(χ)
and let D be a nilpotent derivation on A, then D
(
A
)
is contained in N
(
A
).
In the following lines, we recall definitions and some basic facts about latticeordered algebras. For more information about this field, one can refer to
[1,3]
. A (real) algebra
A
which is simultaneously a vector lattice such that the partial ordering and the multiplication in
A
are compatible, that is
a, b
∈
A
^{+}
implies
ab
∈
A
^{+}
is called
lattice-ordered algebra
( briefly ℓ-algebra). The ℓ-algebra
A
is said to be a
d
-algebra whenever
a
Λ
b
= 0 in
A
implies
ac
Λ
bc
=
ca
Λ
cb
= 0, for all 0 ≤
c
∈
A
. In general,
d
-algebras are not commutative, see
[3]
.
Since any Archimedean
d
-algebra satisfies the condition (χ) ; see [3, Corollary 5.7], we deduce the following result:
Corollary 2.10.
Let A be an Archimedean d-algebra and let D be an algebraic nil derivation on A.Then D
(
A
)
is contained in N
(
A
) :
Definition 2.11.
Let A be an algebra. For a fixed a ∈ A, define D : A → A by D(
x
) = [
x, a
] =
xa
−
ax
, for all
x
∈
A
. Then
D
is called
inner derivation
of
A
associated with
a
and is generally denoted by
D_{a}
.
Theorem 2.12.
Let A be an Archimedean d-algebra with the condition Z
(
A
) = {0},
where Z
(
A
)
denotes the center of A and let D be an inner derivation on A
.
Then the following assertions are equivalent
:
i) D is nilpotent; ii) D ^{3} = 0; iii) D is induced by a nilpotent element .
Proof
. i) ) ii) Let
a
∈
A
such that
D
=
D_{a}
. Since any Archimedean
d
-algebra satisfies the condition (χ) ; then for all
, we have
for all
x
∈
A
. Since
D_{a}
is nilpotent, there exists
n
∈ ℕ such that
Therefore
for all
x
∈
A
. Consequently
a^{2n+1}
∈
Z
(
A
) = {0} : Hence
a^{2n+1}
= 0. By [3, Theorem 5.5], we deduce that
a
^{3}
= 0: It follows that
ii) ⇒ iii)
means that
a
^{3}
= 0. Therefore
a
∈
N
(
A
).
iii) ⇒ i) This path is obvious.□
Remark 2.13.
It is obvious that algebraic nil derivations are nil derivations. The simple-minded attempt to extend Theorem 1,2 and 3 to nil derivations obviously fails. This is illustrated in the following example.
Example 2.14.
Let
A
= ℂ[
X
]
and D
:
A
→
A defined by
It is not hard to prove that D is a nil derivation but not an algebraic nil derivation, whereas
D
(
A
) =
A
≠
N
(
A
).

1. INTRODUCTION

Let
2. THE MAIN RESULTS

To prove our first theorem, we shall need the following algebraic result.
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Acknowledgements

The referees have reviewed the paper very carefully. The author expresses his deep thanks for the comments.

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Citing 'THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS
'

@article{ SHGHCX_2013_v20n4_243}
,title={THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS}
,volume={4}
, url={http://dx.doi.org/10.7468/jksmeb.2013.20.4.243}, DOI={10.7468/jksmeb.2013.20.4.243}
, number= {4}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={TOUMI, MOHAMED ALI}
, year={2013}
, month={Oct}