ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 13 Naude, Cornelia, University of South Africa Asymptotic behaviour of attached primes Let R be a commutative ring with identity and I an ideal of R. Let A be an Artinian R module. Ratliff has shown that the sequence Ass(R/\overline{I^{n}}) is constant for large n, where the bar denotes ordinary integral closure. Sharp and Taherizadeh provided Artinian analogues for the theory of integral closures by defining the integral closure of an ideal relative to an Artinian module A, denoted by I^{∗} . They formulated the following open questions that are dual to Raliff’s results:

(i) Is it the case that Att(0 :_{A} (I^{i})^{∗}) ⊆ Att(0 :_{A} (I^{i+1})^{∗}) for all i ∈ N? (ii) Is the sequence of sets (Att(0 :_{A} (I^{n})^{∗}))_{n∈N} ultimately constant? In this lecture the closure of an ideal I with respect to a submodule N of a Noetherian module M is defined by means of an integer valued filtration on the ring R. An Artinian anologue is also defined, namely the closure Ī_{k,B} of I^{k} relative to a submodule B of an Artinian module A. The main result of this lecture is a positive answer to the question above, with Att(0 :_{A} (I^{i})^{∗}) replaced by Att(B :_{A} Ī_{i,B}) .