Gamma ideal extension in gamma systems

Let γ be a nonempty set, be a commutative γ-semigroup contains γ-identity and M a unitary right -system (act). The notion of a γ-ideal extension on gamma systems are given and studied. An -system M is said to be a γ-ideal extension, if for each -homomorphism f : L → M such that , for each γ-ideal L of . A lot of characterizations and properties of gamma ideal extension of gamma systems have been given. In truth, we offer that γ-ideal extension of -system M is equal to the following similar terms 1. . 2. For all and implies that . 3. For all and . 4. For all , then , for some γ-compatible ρ on .