Asymptotic Primes and Asymptotic Grades on Modules Original Research Article

In this paper we define image*(I, M), image*(I, M), Q(I, M), E(I, M), asymptotic (resp. essential) sequences, asymptotic (resp. essential) grades, and locally quasi-unmixed (resp. locally unmixed) modules for Noetherian modules as counterparts of those for Noetherian rings and it is shown that all the results concerning these for Noetherian rings have valid analogues for modules. Among these are image*(I, M) subset of or equal to image*(I, M) ∩ Q(I, M); image*(I, M) union or logical sum Q(I, M) subset of or equal to E(I, M) subset of or equal to A* (I, M); all the sets are finite; Q(I, M) for general Noetherian modules behaves nicely; and a characterization of locally quasi-unmixed (resp. locally unmixed) modules in terms of asymptotic (resp. essential) sequences.