SOME RESULTS ON NOETHERIAN SEMIGROUP

In this paper we study some results on Noetherian semigroups. We show that if SS is an strongly faithful S-act and S is a duo weakly Noetherian, then we have the following. (1) If A is a finitely generated proper ideal in S, then Tn2N An = (0). (2) If (0) is not a prime ideal of S, then every principal prime ideal of S is a minimal prime ideal of S. Also, if a duo semigroup S has acc on prime ideals and each prime ideal minimal over a proper subideal is finitely generated, then S is Noetherian.