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. Let 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. To prove our first theorem, we shall need the following algebraic result. 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 Dⁿ ( x ), Dⁿ ( x ² ), Dⁿ ( xⁿ ) ∈ N ( A ), where N ( A ) denotes the set of all nilpotent elements of A . Then D ( x ) ∈ N ( A ). Proof . Let x ∈ A with Dⁿ ( x ), Dⁿ ( x ² ), Dⁿ ( xⁿ ) ∈ N ( A ). It follows that Since Dⁿ ( x ) ∈ N ( A ) ; we have Moreover, letting such that n₁ + n₂ = n , this one has By using the Leibnitz rule for D^k ( xⁿ¹ ) and D^k ( xⁿ² ) 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 Dⁿ ( x ) = Dⁿ ( x ² ) = Dⁿ ( xⁿ ) = 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 ( a₁, b₁ ) ( a₂, b₂ ) = ( a₁a₂ + a₂a₁, b₁b₂ + b₂b₁ ) for all ( a₁, b₁ ), ( a₂, b₂ ) 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 Dⁿ ( x ) = Dⁿ ( x ² ) = Dⁿ ( xⁿ ) = 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 Dⁿ ( 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) D³ = 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 a²ⁿ⁺¹ ∈ Z ( A ) = {0} : Hence a²ⁿ⁺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 ).