Introduction

Nobusawa introduced the first Γ-ring which is generalization of the classical concept of a ring. This idea was later marginally undermined by Barnes, with the important algebraic structure being maintained [1-3]. A Γ- ring is in this sense a pairing of additive abelian groups M and Γ along with a mapping M × Γ × M →M to M, (s, γ t) defined by (s, γ, t) ↦ s γ t, satisfying suitable distributive laws and the associativity condition (s γ t) β r = s γ (t β r) and and , for all s, t, r ∈ M and γ, β ∈ Γ [4-6]. A Γ-ring M is called 2torsion free if 2s = 0 implies s = 0. Moreover, M is called prime if s Γ M t = 0 implies, s = 0 or t = 0. It is called semi-prime if s Γ M s = 0 implies s = 0. Every ring is a Γ -ring, of course, but not all rings are necessarily Γ -rings [7-9].

The concept of derivations is significant in the theory of structure of rings and Γ-rings. To generalize the classical theory of derivations in rings, Jing defined derivations and Jordan derivations on Γ-rings [10-12]. Subsequently, a number of authors investigated different generalizations including higher derivations and Jordan higher derivations [13-15]. Chang coined the theory of derivations in rings and examined their algebraic characteristics [16-18]. Later workers generalized these concepts to Γ-rings and studied their behavior in various eponymous conditions. The concept of reverse derivations on rings, introduced by Bresar and Vukman, gave a dual variant of classical derivations and led to the new perspectives in the theory of derivations [7]. It was later developed by Majeed and Salih, who added higher derivations and Jordan higher derivations on Γ-rings along with a number of generalizations and structural properties [8,9]. Having these developments in mind, we generalize the idea of reverse derivations to semi-prime -negative Γ-rings [18-20]. Precisely, we construct higher reverse Γ-rings derivations, Jordan higher reverse Γ-rings derivations and Jordan triple higher reverse Γ-rings M derivations. In this paper, M refers to Semi prime Γ-ring (a 2 torsion free).

2. Reverse Alpha- derivation of Γ-ring M

2.1 Definition

Figure 1.

Let us show examples of reverse a-derivation on G-ring .

2.2. Example

Assume R a ring, the additive abelian groups

Figure 2.

2.3 Example

Figure 3.

3. Higher Reverse -Derivation of -ring M

Let us begin the following concept:

3.1 Definition

Figure 4.

Let us notice an example on higher reverse Alpha-derivation of Γ-ring.

3.2 Example

Figure 5.

4. Key Marks

In this part we will current and learning some properties of upper reverse -derivation on -ring M which make us able to prove several theorem concerning this definition as orthogonally.

4.1 Lemma

Suppose Alpha = (Alpha_i)_{i eN} collection of endomorphism of Γ-ring M, d = (d_i)_{i eN} Jordan higher reverse Alpha-Derivations on M . Then the resulting statements hold :

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Conclusion

The concept of higher reverse 2-torsion free semi-prime α-derivation Γ -rings was explored in this paper. The paper generalizes the classical theory of derivations and higher derivations in rings into a more general allegorical structure. The higher reverse α-derivations, Jordan higher reverse α-derivations and higher reverse α-derivations were defined. These concepts provide natural generalizations of reverse derivations and allow a deeper understanding of additive mappings in Γ-rings. Several structural properties of these mappings were established. In particular, we derived identities and relations that describe the behavior of Jordan higher reverse -derivations and their interaction with the algebraic structure of semi-prime Γ-rings. These findings show that the setting of Γ-rings can have many of the same qualities as derivations in classical ring theory.

The obtained results contribute to the development of derivation theory in generalized algebraic systems and open the possibility for further research on generalized derivations, higher mappings, and related operator structures in Γ-rings and other algebraic frameworks.

Future work may focus on studying these derivations on prime Γ-rings, with involution, or investigating their connections with generalized derivations and automorphism structures.