What is it about?
Let (R,m) be a Noetherian local ring and (S,n) denote the m-adic completion of R. In this paper it is shown that the following conditions are eqivalent: 1) the going-up theorem holds between R and S. 2) for each ideal J of R and each non-zero Artinian R-module M, the J-cofiniteness of M is equivalent to this condition that the ideal (Ann M + J) of R is m-primary.
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Why is it important?
This result provides an easy criterion for detecting that when the going-up theorem holds between a Noetherian local ring and its completion.
Perspectives
This paper was an improvement of the main results of the earlier published article: N. Abazari, and K. Bahmanpour, "A note on the Artinian cofinite modules", Commun. Algebra, Vol(42) No: 3 (2014), pp. 1270-1275. I hope that you find this article enough useful for your future research. Also, on behalf of the authors, I should apologize from the readers for our mistakes in typing the symbols at the form (0:_{M} J), because of the displacement of M and J in several places.
Dr Kamal Bahmanpour
bahmanpour.k@gmail.com
Read the Original
This page is a summary of: Artinian Cofinite Modules and Going-up for R ⊆ R ̂ $R\subseteq \widehat {R}$, Acta Mathematica Vietnamica, March 2017, Springer Science + Business Media,
DOI: 10.1007/s40306-017-0203-6.
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