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(Generalized) filter properties of the amalgamated algebra

Yusof Azimi (2022)

Archivum Mathematicum

Let R and S be commutative rings with unity, f : R → S a ring homomorphism and J an ideal of S . Then the subring R ⋈ f J : = { ( a , f ( a ) + j ) ∣ a ∈ R and j ∈ J } of R × S is called the amalgamation of R with S along J with respect to f . In this paper, we determine when R ⋈ f J is a (generalized) filter ring.

k -torsionless modules with finite Gorenstein dimension

Maryam Salimi, Elham Tavasoli, Siamak Yassemi (2012)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring. It is shown that the finitely generated R -module M with finite Gorenstein dimension is reflexive if and only if M 𝔭 is reflexive for 𝔭 ∈ Spec ( R ) with depth ( R 𝔭 ) ≤ 1 , and G- dim R 𝔭 ( M 𝔭 ) ≤ depth ( R 𝔭 ) - 2 for 𝔭 ∈ Spec ( R ) with depth ( R 𝔭 ) ≥ 2 . This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n ≥ 2 we give a characterization of n -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...

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