Open AccessThe asymptotic closure of an ideal relative to a module
Author(s) -
Sylvia M. Foster,
Johnny A. Johnson
Publication year - 2001
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.32.2001.379
Subject(s) - mathematics , ideal (ethics) , noetherian ring , unitary state , noetherian , closure (psychology) , commutative ring , primary ideal , pure mathematics , ring (chemistry) , identity (music) , maximal ideal , commutative property , discrete mathematics , algebra over a field , principal ideal ring , law , chemistry , physics , organic chemistry , political science , acoustics
In this paper we introduce the concept of the asymptotic closure of an ideal of a commutative ring $ R $ with identity relative to a unitary $ R $-module $ M $. This work extends results from P. Samuel, M. Nagata, J. W. Petro and Sharp, Tiras, and Yassi. Our objectives in this paper are to establish the cancellation law for the asymptotic completion of an ideal relative to a finitely generated module and show that the integral closure of an ideal relative to a Noetherian module $ M $ coincides with the asymptotic closure of the ideal relative to the Noetherian module $ M $