Artinian Cofinite Modules and Going-up for R⊆R̂

• Published in: Acta mathematica vietnamica Vol. 42; no. 4; pp. 605 - 613
• Main Authors: Pirmohammadi, Gholamreza, Ahmadi Amoli, Khadijeh, Bahmanpour, Kamal
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 2017; Springer Nature B.V
• Subjects: Arches; Equivalence; Mathematics; Mathematics and Statistics; 13E05; 14B15; Attached prime; Cofinite module; Noetherian ring; 13D45; Local cohomology
• ISSN: 0251-4184; 2315-4144
• DOI: 10.1007/s40306-017-0203-6

Let ( R , 𝔪 ) be a commutative Noetherian local ring. In this paper, it is shown that the going-up theorem holds for R ⊆ R ̂ if and only if Rad ( I + Ann R A ) = 𝔪 for any proper ideal I of R and any non-zero Artinian I -cofinite module A . Furthermore, using the main result of Zöschinger, Arch. Math. 95 , 225–231 ( 2010 ), it is shown that these equivalent conditions are equivalent to R being formal catenary with α ( R ) = 0 and to Att R H I dim M ( M ) = { 𝔭 ∈ Assh R ( M ) : Rad ( 𝔭 + I ) = 𝔪 } for any ideal I of R and any non-zero finitely generated R -module M .