In this paper we introduce the notion of prime one-sided ideal in S where (S, Γ) is a Γ- semigroup and show that there is a one-to-one correspondence between the set of all prime right ideals of S and the set of all prime right ideals of M where M i...
In this paper we introduce the notion of prime one-sided ideal in S where (S, Γ) is a Γ- semigroup and show that there is a one-to-one correspondence between the set of all prime right ideals of S and the set of all prime right ideals of M where M is the left operator semigroup of the Γ-semigroup (S, Γ). We also introduce the notions of right Noetherian on S where (S, Γ) is a¡-semigroup and Noetherian Γ-semigroup and study it. Lastly we show that a commutative Γ-semigroup (S, Γ) with unities is Noetherian if and only if every. prime ideal of S is fnitely generated, thus extending a result of Cohen's Theorem [1] for semigroup toΓ-semigroup.