Rees algebra over an ideal in tt-rings

If there exists a finitely generated module M   over a tt-ring such that tM=AnnMttM=AnnMt and M/tMM/tM is of finite length, then we show that commutator ideal [q,q][q,q] is contained in tqtq, for some prime ideal q under certain conditions. Using this property, we discuss the Noetherian property of the Rees algebra of the prime ideal q   in tt-ring and as an application we give an analogue of the Artin–Rees lemma for tt-rings.