Asymptotic Primes and Asymptotic Grades on Modules

✍ Scribed by S.H. Ahn

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 973 KB
• Volume: 174
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1163

✦ Synopsis

In this paper we define (\overline{\mathbf{Q}}^{}(I, M), \overline{\mathbf{A}}^{}(I, M), \mathbf{Q}(I, M), \mathbf{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 (\overline{\mathbf{Q}}^{}(l, M) \subseteq) (\overline{\mathbf{A}}^{}(I, M) \cap \mathbf{Q}(I, M) ; \overline{\mathbf{A}}^{}(I, M) \cup \mathbf{Q}(I, M) \subseteq \mathbf{E}(I, M) \subseteq \mathbf{A}^{}(I, M)); all the sets are finite; (\mathbf{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. (C) 1995 Academic Press, Inc.